On Jacobians of dimension 2g that decompose into...

On Jacobians of dimension 2g that decompose into Jacobians of

dimension g

by

Avinash Kulkarni

B.Math (Hons.),University of Waterloo, 2012

Thesis Submitted in Partial Fulfillment

of the Requirements for the Degree of

Master of Science

in the

Department of Mathematics

Faculty of Science

c© Avinash Kulkarni 2014

SIMON FRASER UNIVERSITY

Summer 2014

All rights reserved.

However, in accordance with the Copyright Act of Canada, this work may be

reproduced without authorization under the conditions for “Fair Dealing.”

Therefore, limited reproduction of this work for the purposes of private study,

research, criticism, review and news reporting is likely to be in accordance

with the law, particularly if cited appropriately.

APPROVAL

Name: Avinash Kulkarni

Degree: Master of Science

Title of Thesis: On Jacobians of dimension 2g that decompose into Jacobians of

dimension g.

Examining Committee: Dr. Ralf Wittenberg, Associate Professor

Chair

Dr. Nils Bruin

Senior Supervisor

Associate Professor

Dr. Imin Chen

Supervisor

Associate Professor

Dr. Petr Lisonek

Internal Examiner

Associate Professor

Date Defended/Approved: August 12th, 2014

ii

Partial Copyright Licence

iii

Abstract

In this thesis we describe a family of Jacobian varieties of non-hyperelliptic genus 2g curves that are

isogenous to a product of Jacobians of genus g curves in a specific way. For any hyperelliptic genus

g curve C we construct a 2-parameter family of hyperelliptic genus g curves H with J(H)[2] iso-

morphic to J(C)[2], and a generically non-hyperelliptic curve A such that there is an isogeny from

J(C) × J(H) to J(A) whose kernel is the graph of the isomorphism taking J(H)[2] to J(C)[2].

This is accomplished by first showing that C can be considered as a subcover of a Galois cover

of a P1 that has A and H naturally arising as subcovers and then showing the naturally occurring

isogeny relations have the desired kernel. We also list some corollaries to the main result and pro-

vide a magma script to generate non-hyperelliptic genus 4 curves that have curious automorphism

groups.

Keywords: Jacobian variety; decomposition

iv

Acknowledgements

I would like to thank my family for the many years they have supported and encouraged my study

of mathematics. To my fellow students Navid Alaei, Steve Melczer, Ryan McMahon, and Brett

Nasserden I thank you for the many good times and for the helpful advice in learning the nuances of

algebraic geometry. I owe many thanks to Colin Weir for his helpful discussions and insight into my

research project, particularly regarding possible generalizations of the main result. Thanks to Imin

Chen for pointing out the connection to Kani and Rosen’s work. I would like to thank Bjorn Poonen

for his very insightful comments regarding the future directions section. I would like to thank Diane

Pogue for helping me with the administrative aspects of the writing process. Finally, I am deeply

grateful to my senior supervisor Nils Bruin for his enduring patience, wise advice, vigilant editing,

and warm guidance.

v

Contents

Approval ii

Partial Copyright License iii

Abstract iv

Acknowledgements v

Contents vi

List of Figures viii

1 Introduction 1

2 Background material 42.1 Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Notation and persistent assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Theorems regarding curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 Riemann-Roch and Riemann-Hurwitz Theorems . . . . . . . . . . . . . . 13

2.4.2 Computing ramification . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Definition and properties of abelian varieties . . . . . . . . . . . . . . . . 19

2.5.2 Definition and properties of the Jacobian . . . . . . . . . . . . . . . . . . 25

2.5.3 Polarizations, principal polarizations, and polarized isogenies . . . . . . . 25

2.5.4 Decompositions of the Jacobian . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Endomorphisms of abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . 29

vi

2.7 Final preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7.1 Motivating facts for the case g = 2 . . . . . . . . . . . . . . . . . . . . . . 32

2.7.2 Representing 2-torsion points on hyperelliptic Jacobians . . . . . . . . . . 33

3 Curves of genus 2g with decomposable Jacobians 353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Construction 1: Legendre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Diagrams associated to a Galois covering . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Construction of a dihedral cover of P1k . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.1 Norm construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Verifying J(CF )× J(Cf )/∆ ∼= J(A) . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.1 Computing with endomorphisms of J(Ω) . . . . . . . . . . . . . . . . . . 49

3.5.2 Kernel data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.7 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.7.1 MAGMA script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Future directions 584.1 Converse to the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 Other future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Bibliography 62

Appendix A MAGMA script 64A.1 "Elliptic_Decomposition.m" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

vii

List of Figures

3.1 Subcover structure of Ω/P1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Subgroup structure of Aut(Ω/P1). Arrows denote inclusion. . . . . . . . . . . . . 41

3.3 Subcover structure of Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

viii

Chapter 1

Introduction

An abelian variety is a projective variety together with a morphism +: A × A → A and a distin-

guished point O such that its point set is a commutative group with the operation given by "+".

The group structure on an abelian variety can sometimes be used to recover arithmetic information

about the subvarieties that exist within it. Considering curves inside their Jacobian varieties lead to

Faltings’ famous (revised) proof of Mordell’s conjecture [10, Section E.1].

Theorem 1.0.1 (Faltings). A curve of genus g ≥ 2 defined over a number field k has finitely many

k-rational points.

There are a number of ways in which abelian varieties can decompose into a product of abelian

varieties of smaller dimension. It can be a product itself or it can admit a finite morphism onto such a

product. Such a morphism is called an "isogeny". The factors of a decomposable abelian variety can

be analyzed to understand the original object much like other algebraic structures. Decomposability

of abelian varieties has a long history in mathematics that goes back at least to the computation of

abelian and elliptic integrals in the late 19th century [1, 17].

The modern study of the subject has led to a number of interesting geometric and arithmetic

results. There is a large body of work finding g-dimensional Jacobian varieties that are isogenous

to the product of g copies of one elliptic curve. Jacobians of this type are interesting for a number

of reasons, one reason of interest to number theorists and cryptographers is that over a finite field

Jacobian varieties of this type are closely related to Jacobians that have a maximal number of rational

points [11]. A paper by Ekedahl and Serre [6] shows that Jacobians of this type exist for many values

of g. Related work by Jennifer Paulhus [16] classifies the other isogeny types of Jacobian varieties

1

CHAPTER 1. INTRODUCTION 2

of small genus. Her work also has applications to the computation of rank bounds on elliptic curves,

which is a topic of much interest in modern mathematics.

All abelian varieties can be assigned something called a polarization, which is not in general

preserved by isogeny. It is a natural and useful question to ask when a decomposition does respect

the polarization. Nils Bruin and Victor Flynn give an example of how such a decomposition can aid

in determining the existence of rational points on curves of genus ≤ 2, see [2, 3].

In this thesis we show how an explicitly described family of hyperelliptic curves can be related

to Jacobian varieties that decompose in a way that respects polarizations. We prove

Theorem (Main Result). Let k be a field of characteristic not equal to 2. Let Cf be a hyperelliptic

genus g curve defined over k, and J(Cf ) its Jacobian. Then there exists a two parameter family of

explicitly determined curves CF of genus g and A of genus 2g such that

1. CF is hyperelliptic and there is an isomorphism of finite algebraic sets

ψ : J(CF )[2]→ J(Cf )[2].

2. A is a double cover of CF .

3. J(A) ∼= J(Cf )× J(CF )/∆ as polarized abelian varieties, where ∆ is the (anti)-diagonally

embedded 2-torsion of J(Cf ).

Our work was largely inspired by Everett Howe’s [11, Section 4] classification of genus 4 double

covers of genus 2 curves with a rational point since the decomposition type studied in this thesis

arises when a genus 4 curve is a double cover of a genus 2 curve. We remark that any genus 4 curve

that can be constructed from Howe’s technique can also be produced from our construction with the

right choice of Cf and µ but not vice-versa. We differ from [16] since we allow non-elliptic factors

in the decomposition but we restrict the kernel of the isogeny from the product variety. We also

draw inspiration from the construction of Legendre [1]. Our construction generalizes [1] since we

do not require curves with a rational Weierstrass point. The techniques used in this thesis have also

been applied by Recillas [18] and Donagi [4] to find correspondences between Jacobian varieties

and Prym varieties.

In Chapter 2 of this thesis we provide an exposition of the necessary language required to state

and prove the main result. In Chapter 3 we prove the main result and then state some immediate

consequences. We also provide a small magma script that constructs one of the decomposition types

CHAPTER 1. INTRODUCTION 3

in [16] explicitly. Finally in Chapter 4 we posit a future statement related to the main result that can

serve as a future direction of research.

Chapter 2

Background material

2.1 Prelude

In this chapter we present a terse review to the arithmetic geometry of curves and their Jacobians.

This chapter shall serve the purpose of refreshing the reader on the definitions. The chapter as a

whole serves to exposit on the language of arithmetic geometry to a point where the main question

can be well formulated as well as provide the necessary tools needed to prove it. Those interested in

the details are encouraged to refer to [9, 10, 19, 20], and Milne’s course notes [14].

2.2 Notation and persistent assumptions

This section serves as a shorthand glossary and establishes the conventions and notations in case the

reader should want to refer back to it.

If X is a set with finite cardinality then we denote the number of elements by #X or by |X|.The Klein 4-group, which is isomorphic to Z/2Z × Z/2Z, is denoted by V4. The dihedral group

of order 8 is denoted by D4. We let k denote an arbitrary field with characteristic not 2. We shall

always denote its algebraic closure as k.

We define affine n-space over k, denoted Ank , to be the set of all n-tuples of elements of k.

An element P ∈ Ank will be called a point. We define projective n-space over k, denoted Pnk to

be the set of (n + 1)-tuples of k, excluding the all-zero tuple, modulo the relation (a0, . . . , an) ∼(λa0, . . . , λan), λ ∈ k∗. A point P ∈ Pnk is one of these equivalence classes and is denoted P =

(a0 : . . . : an).

4

CHAPTER 2. BACKGROUND MATERIAL 5

For polynomials fi ∈ k[x1, . . . , xn] we let V (f1, . . . , fr) ⊆ Ank

denote their common zero locus

and call this an affine algebraic set defined over k. Similarly we let V (f1, . . . , fr) ⊆ Pnk

denote the

common zero locus of homogeneous polynomials fi with coefficients in k and call this a projective

algebraic set defined over k. Any affine or projective algebraic set defined over k is also defined

over k. As a shorthand we emphasize that an affine algebraic set X ⊆ Ank

is defined over k by

writing X ⊆ Ank and we use a similar shorthand for projective algebraic sets. Henceforth if we

make a statement regarding affine algebraic sets that has an analogue for projective algebraic sets

we shall use the term algebraic set. We denote the set of points of an algebraic set X by X(k).

An algebraic setX defined over k is said to be geometrically reducible if we can find non-empty

algebraic sets Y,Z defined over k such that X = Y ∪ Z and both Y 6⊆ Z,Z 6⊆ Y . Otherwise X

is said to be geometrically irreducible or more commonly, we call X a variety. Since every variety

in this thesis is either an affine variety or a projective variety we identify a variety with its point set

over k.

Let X and Y be varieties defined over k. Fix a choice of co-ordinates x1, . . . , xn for X and

y1, . . . yr for Y . We may describe any morphism ϕ : X → Y by polynomial functions f1, . . . , fr in

the co-ordinates ofX . A morphism of varieties defined over k is a morphismϕ := (f1, . . . , fr) : X →Y such that each fi is a polynomial with coefficients in k. Similarly, a rational map defined over k

is a map ϕ := (f1, . . . , fr) : X → Y such that each fi is a rational function that has coefficients in

k. A rational map φ : X → Y is said to be dominant if there is are open sets UX ⊆ X and UY ⊆ Ysuch that φ(UX) = UY . A function on a variety X defined over k is a rational map ϕ : X → A1

k

defined over k. The ring of rational functions on X defined over k, also called the function field of

X , is denoted by k(X). We also call k(X) the function field of X . We denote the identity map on

X by 1X and when it is clear from context we will drop the subscript. The identity map is always

defined over k.

The absolute galois group Gk := Gal(k/k) is the group of automorphisms of k that fix k. Let

X be a projective variety defined over k and let P := (w0 : . . . : wn) ∈ X(k). We say P is a

k-rational point if there is a λ ∈ k such that each λwi ∈ k. If X is an affine variety defined over k

and P := (w0, . . . , wn) ∈ X(k) we say P is a rational point if each wi ∈ k. The rational points of

a variety X defined over k are denoted X(k).

We say that a curve defined over k is a birational isomorphism (defined over k) class of varieties

defined over k of dimension 1. We call a particular representative a model of a curve. We call a


model projective or affine if the representing variety is projective or affine respectively. Theorem

2.3.9 shows that any such class contains smooth projective models and any two such models are

isomorphic. Therefore we will often identify a curve with its smooth model.

Let X and Y be projective curves defined over k and let φ : Y → X be a surjective morphism

of curves defined over k. There is a corresponding morphism of function fields φ∗ : k(X) → k(Y )

given by φ∗(f) := f φ.

Definition 2.2.1. Let φ : X → Y be a surjective morphism of curves defined over k. If

[k(Y ) : φ∗k(X)] is a finite extension of fields then we call this quantity the degree of φ and φ is said

to be separable if [k(Y ) : φ∗k(X)] is separable.

Let φ : C1 → C2 be a surjective morphism of models of curves and let P ∈ C2(k). We shall see

by a later result (Proposition 2.3.6) this automatically ensures φ is of finite degree. Then we call the

set φ−1(P ) the fibre over P . We also say that C1 is a cover of C2 and that φ is the covering map. A

double cover is a cover of degree 2.

We denote the group of automorphisms of a variety X by Aut(X). If π : C → C is a non-

constant morphism of curves then we denote the subgroup of automorphisms σ ∈ Aut(C) such that

π σ = π by Aut(C/C). An involution of a variety X is an automorphism µ such that µ µ = 1X .

2.3 Theorems regarding curves

In this section when we refer to a point on a curve we mean P ∈ C(k). We shall also assume that

every curve is defined over k.

Definition 2.3.1. Let P ∈ C be a point. Then the local ring at P is defined by

OC,P := f ∈ k(C) : ∃U ⊆ C Zariski open such that P ∈ U and f is regular on U .

Proposition 2.3.2. If P is a smooth point of C then OC,P is a discrete valuation ring.

Proof. See [20, Proposition II.1.1].

Recall that a generator for the unique maximal ideal of a discrete valuation ring is called a

uniformizing parameter or a uniformizer.


Definition 2.3.3. Let f ∈ k(C) and let t be a uniformizing parameter at P . We define

ordP f := supd ∈ Z : f · t−d ∈ OC,P

.

Definition 2.3.4. Let f ∈ k(C) be a rational function of C. A zero of f is a point P such that

ordP f > 0. Similarly a pole of f is a point P such that ordP f < 0.

Proposition 2.3.5. Let C be a non-singular projective model of a curve. Then any f ∈ k(C) has

finitely many poles and zeros. Moreover∑

P∈C ordP (f) = 0.

Proof. See [20, Proposition II.1.2] for the first statement and [20, Proposition II.3.1] for the second.

We conclude this section with some general theorems about curves which will come in handy later.

Proposition 2.3.6. Let π : C → C be a non-constant morphism of curves. Then π is surjective and

of finite degree.

Proof. See [20, Theorems II.2.3, II.2.4].

Definition 2.3.7. If π : C → C is a surjective morphism of curves then we refer to C as a cover of

C.

Proposition 2.3.8. Let φ : C → C ′ be a birational map. Then k(C) ∼= k(C ′).

Proof. See [20, Theorem II.2.4].

Theorem 2.3.9. Let C be a curve.

1. Then there is a smooth projective curve X such that X is birationally equivalent to C.

2. If X and X ′ are smooth projective curves birationally equivalent to C then X ′ is isomorphic

to X .

We say X is a desingularization of C.

Proof. See [8, 7.5 Theorem 3].

Proposition 2.3.10. Let φ : C1 → C2 be a rational map of projective curves and let C1 be smooth.

Then φ can be extended to a morphism on all of C1.



Corollary 2.3.11. Let C1, C2 be projective curves and let C1, C2 be their respective desingulariza-

tions. If φ : C1 → C2 is a birational morphism then there is a morphism φ : C1 → C2.

Proof. We notice that by definition C1, C2 are birational to C1, C2 respectively and that by assump-

tion C1 is birational to C2. Hence there is a birational morphism φ : C1 → C2. By the preceeding

proposition φ extends to a morphism.

2.4 Divisors

The following is a very brief treatment of Weil divisors. This is all we need since we only work with

smooth curves. We introduce the Picard group of a curve and survey some useful properties. We

also provide a couple of computational lemmas at the end of the section for use later. The reader

interested in this subject is encouraged to refer to [10] for a more complete reference.

Definition 2.4.1. Let X be a variety. A subvariety Y is said to be of co-dimension 1 if for every

variety Y ⊆ Z ⊆ X we have Z = Y or Z = X .

Definition 2.4.2. Let X be a smooth projective variety. A (Weil) divisor of X is a formal Z-linear

combination D =∑

Y⊆X aY Y such that all but finitely many of the aY are zero and each Y is a

codimension 1 subvariety of X . The free abelian group generated by the Y is denoted Div(X).

In other words, Div(X) is the group of divisors of X over k. In this thesis we exclusively focus

on divisors of curves, however it is possible to extend this notion to arbitrary projective varieties

as in [10]. We observe that since every codimension 1 subvariety of a curve must be a point that

every divisor on a curve can be written as the formal linear combination of k-points on the curve.

Throughout let C be a smooth projective curve.

Definition 2.4.3. Let D =∑

P∈C aPP be a divisor of C. Then the multiplicity of a point P in D is

the integer aP .

Definition 2.4.4. A divisor D =∑

P∈C aPP is said to be effective if each aP ≥ 0.

Definition 2.4.5. For a divisorD =∑

P∈C aPP on a curveX we define the degree to be∑

P∈C aP .

This is denoted deg(D).


Definition 2.4.6. The degree map deg : Div(C)→ Z is a morphism of groups, the kernel of which

are the degree 0 divisors. The subgroup of degree 0 divisors on C is denoted Div0(C).

Definition 2.4.7. Let π : C → C be a cover of degree d of curves defined over k and let P ∈ C.

Then we have an induced inclusion of function fields π∗ : k(C)→ k(C). We define the ramification

index of π at P by

eπ,P := ordP (π∗(t))

where t is a uniformizer of π(P ). We say that π is ramified at P if eπ,P > 1 and unramified at P

otherwise. We say that π is ramified if there is a ramified point P ∈ C and is unramified otherwise.

If the map π is clear from context then we use the notation eP .

Proposition 2.4.8. Let π : C → C be a cover of curves. Then:

(a) For all Q ∈ C we have ∑P∈π−1(Q)

eP = deg(π).

(b) For all but finitely many P we have eP = 1.


By Proposition 2.4.8 (b) we define the following.

Definition 2.4.9. The ramification divisor of a cover π : C → C is

Rπ :=∑P∈C

(eP − 1)P.

Definition 2.4.10. For any function f ∈ k(C) we define

div(f) =∑P∈C

ordP (f) · P.

By Proposition 2.3.5 this is well defined and degree 0. A divisor of the form div f is called a

principal divisor.


Proposition 2.4.11. Let f, g ∈ k(C) and c ∈ k∗. Then since each ordP is a valuation trivial on the

constant functions we have:div(fg) = div(f) + div(g)

div

(1

f

)= −div(f)

div(c) = 0.

The above proposition allows us to give the following definition.

Definition 2.4.12. We denote by Princ(C) the subgroup of principal divisors in Div0(C).

Definition 2.4.13. We define the Picard group Pic(C) by the exact sequence

0 // Princ(C) // Div(C) // Pic(C) // 0

Similarly, define Pic0(C) by the exact sequence

0 // Princ(C) // Div0(C)[·] // Pic0(C) // 0

The group Pic(C) is the divisor class group of C. We represent elements in Pic(C) by [D] and

call this the divisor class of D.

Definition 2.4.14. Let C, C be smooth projective curves and let π : C → C be a cover. Let D =∑P∈C nPP be a divisor of C. We define the pullback of D, denoted π∗(D), as

π∗(D) =∑P∈D

∑π(Q)=P

eQnPQ

where eQ is the ramification index of Q.

Lemma 2.4.15. There is an induced morphism

π∗ : Pic0(C)→ Pic0(C).

Proof. It is straightforward to verify the claim that

π∗(div(f)) = div(f π) = div(π∗(f))


and hence the pullback of a principal divisor is again principal. Hence we get the induced map from

the diagram

0 // Princ(C) //

π∗

Div0(C) //

π∗

Pic0(C) //

0

0 // Princ(C) // Div0(C) // Pic0(C) // 0

Definition 2.4.16. Let π : C → C be a cover of curves. Then we define the map π∗ : Div0(C) →Div0(C) by

π∗

∑P∈C

nPP

=∑P∈C

nPπ(P ).

We call π∗ the norm.

Definition 2.4.17. If σ ∈ Aut(C/C) and D =∑

P∈C aPP is a divisor then we have the group

action

σ∗(D) =∑P∈C

aPσ(P ).

The following appears as an exercise in [9] but proves to be useful to us later.

Lemma 2.4.18. Let C, C, π be as before and let P ∈ C. If C/C is Galois then Gal(C/C) acts

transitively on the set

fibre over P :=Q ∈ C(k) : π(Q) = P

.

Proof. Label the points in the fibre over P by Q1, . . . , Qn. By [19, Theorem III.2.3] we find ti ∈k(C) such that ordQj ti = δij where δ is the Kronecker delta. Since

NmC/C

(t1)(Q) =∏

σ∈Gal(C/C)

t1(σ(Q))

is invariant under Galois action and vanishes atQ1, it must vanish at eachQ. That is, for every point

such that π(Q) = P there is a σ such that t1(σ(Q)) = 0. Since t1 had a unique root among the Q

we are done.

Remark 2.4.19. Our blanket assumption that char k 6= 2 guarantees that a double cover of curves

is always separable.


Lemma 2.4.20. Let π : C → C be a Galois cover of smooth projective curves of degree n. Let D

be a divisor on C and D a divisor on C. Then

(i) (π∗ π∗)(D) = nD

(ii) (π∗ π∗)(D) =∑

σ∈Gal(C/C)

σ(D).

Proof.

(i) Let P ∈ C(k). Then

π∗(P ) =∑

π(Q)=P

eQQ

π∗

∑π(Q)=P

eQQ

=∑

π(Q)=P

eQP = nP.

Now let D = a0P0 + . . .+ arPr ∈ Div(C). Then

π∗π∗(D) = π∗π

∗(a0P0 + . . .+ arPr) = a0π∗π∗(P0) + . . .+ arπ∗π

∗(Pr)

= nD.

(ii) Let Q ∈ C(k) and let π(Q) = P . Then

π∗π∗(Q) =∑

π(Q′)=P

eQ′Q′.

Since C/C is Galois, the automorphisms act transitively on the fibre over P . Thus

∑σ∈Gal(C/C)

σ(Q) =∑

Q′∈Orb(Q)

|Stab(Q)|Q′ =∑

π(Q′)=P

eQ′Q′.

Now let D = a0P0 + . . .+ arPr ∈ Div(C). Applying the same trick as before we obtain the

result.

Lemma 2.4.21. Let π : C → C be a Galois cover of curves of degree n. Then there is an induced


morphism of Picard groups π∗ : Pic0(C)→ Pic0(C) given by

π∗

([∑nPiPi

])=[∑

nPiπ(Pi)].

Proof. First we show that π∗ takes principal divisors to principal divisors. By the morphism π there

is an induced inclusion of function fields π∗. There is also the standard norm map Nm: k(C)∗ →k(C)∗. We claim that

k(C)∗

Nm

div // Princ(C)

π∗

k(C)∗

div // Princ(C)

commutes. First we show this for functions g ∈ π∗(k(C)) ⊆ k(C). We see by Lemma 2.4.20 that

π∗ divC

(g π) = n divC g = divC Nm(g).

Now let g ∈ k(C)∗ and let g = Nm(g). Then since π∗(g) = π∗(gσ) for all σ ∈ Aut(C/C) we

haven · π∗ div

C(g) =

∑σ∈Aut(C/C)

π∗ divCgσ = π∗ div

CNm(g) = n · divC g.

So π∗ of principal divisors are still principal. Now consider the diagram

0 // Princ(C) //

π∗

Div0(C) //

π∗

Pic0(C) //

0

0 // Princ(C) // Div0(C) // Pic0(C) // 0

We get an induced map between Picard groups.

2.4.1 Riemann-Roch and Riemann-Hurwitz Theorems

We introduce the standard results for working with projective curves and in particular also define the

genus of a curve. To that end we introduce a differential on a curve. At the end of this section we

provide some explicit computational tools that allow us to determine the data used in these formulae.

Definition 2.4.22. We define the k(C)-module of Kähler differentials ΩC as the free k(C) module


generated by the symbols df for each f ∈ k(C) modulo the relations:

d(f + g)− df − dg = 0

dfg − fdg − gdf = 0

da = 0 for all a ∈ k.

Proposition 2.4.23. Let C be a curve, let P ∈ C, and let t ∈ k(C) be a unformizer at P.

(a) For every ω ∈ ΩC there exists a unique function g ∈ k(C), depending on ω and t, satisfying

ω = g · dt.

We denote g by ω/dt.

(b) Let ω ∈ ΩC with ω 6= 0. The quantity

ordP (ω/dt)

depends only on ω and P , independent of the choice of uniformizer t.


This motivates the following definition:

Definition 2.4.24. Let ω be a differential on a smooth projective curve C, let P ∈ C, and t be a

uniformizer at P . Then we define

ordP (ω) := ordP (ω/dt).

Proposition 2.4.25. For a non-zero ω ∈ ΩC we have ordP ω = 0 for all but finitely many P .


Definition 2.4.26. Let ω be a differential. Then its divisor is defined by

div(ω) :=∑P∈C

ordP (ω)P

which is a divisor by the above proposition. We call this a canonical divisor.


Remark 2.4.27. Since any two non-trivial differentials are k(C)-multiples of each other we see

that their divisors are all linearly equivalent. Thus we see that for non-zero ω ∈ ΩC we have that

[divω] = κ ∈ Pic(C) is independent of ω. We call κ the canonical divisor class of C.

We now finally approach the Riemann-Roch Theorem.

Definition 2.4.28. To a divisor D on a curve X we associate a k-vector space called the Riemann-

Roch space of D defined by

L(D) =f ∈ k(X) : div f +D is effective

∪ 0 .

Proposition 2.4.29. L(D) is a finite dimensional k-vector space.

Proof. See [20, Proposition II.5.2b].

Definition 2.4.30. For notational convenience we define

`(D) := dimk L(D).

Remark 2.4.31. Let D be a divisor on C and let f be a function. Then as vector spaces

L(D) ∼= L(D + div f).

We mention this to point out that the statement of the next theorem is independent of the choice

of canonical divisor.

Theorem 2.4.32 (Riemann-Roch). Let D be a divisor on C and let κ denote a canonical divisor.

Let `(D) = dimL(D). Then there exists an integer g ≥ 0 depending only on C such that

`(D)− `(κ−D) = deg(D)− g + 1.

Proof. See [20, Theorem II.5.4].

Definition 2.4.33. For a given C the integer g in the above theorem is called the genus of C.

By substituting D = 0 and then D = κ into the above theorem and observing that L(0) is the k

vector space of constant functions we obtain:


Proposition 2.4.34. For `, g, k as above,

`(κ) = g

deg(κ) = 2g − 2.

Theorem 2.4.35 (Riemann-Hurwitz). Let π : C → C be a cover of degree d such that the extension

of function fields k(C)/k(C) is separable and let eP be the ramification index of P ∈ C. Then

κC

= π∗κC +Rπ.

If in addition we know char(k) - eP for each P or char(k) = 0 then we take degrees to see

2g(C)− 2 = d(2g(C)− 2) +∑P∈C

(eP − 1).

Proof. See [20, Theorem II.5.9] and [10, Proposition A.2.2.8].

2.4.2 Computing ramification

Computing ramification data from the function fields

In order to make use of the Hurwitz formula we shall require information about the ramification

divisor. This section highlights a means to obtain this.

Definition 2.4.36. Let F be a field. A discrete valuation on F is a map ν : F → Z ∪ +∞ such

that

(i) ν(a) = +∞ if and only if a = 0,

(ii) ν(ab) = ν(a) + ν(b),

(iii) ν(a+ b) ≥ min(ν(a), ν(b)), and

(iv) there exists an element t ∈ F ∗ such that ν(t) = 1.

The pair (F, ν) is called a discrete valuation field.

Let C be a smooth projective curve and k(C) its function field. Let P ∈ C be a point and


OC,P ⊆ k(C) the associated local ring. This gives rise to a discrete valuation on k(C) by defining

νP (f) := ordP f.

In the form of a proposition:

Proposition 2.4.37. To each point P ∈ C(k) on a smooth projective curve we can associate a

discrete valuation νP of k(C) with νP (a) = 0 for all a ∈ k. Additionally, we see that the associated

discrete valuation ring OC,P contains k and the fraction field of OC,P is k(C).

Proposition 2.4.38. Let C be a smooth projective curve and let Oν be a discrete valuation subring

of k(C) containing k such that the fraction field of Oν is k(C). Then there is a point P ∈ C(k)

such that ν = ordP f .

Proof. See [8, Corollary 7.1.4].

The main advantage of this is that we can compute ramification of a cover of curves directly

from the associated function fields. We state a well known result which allows us to easily compute

the ramified places of a separable double cover.

Corollary 2.4.39. Let π : C → C be a separable cover of smooth projective curves and let k(C) =

k(C)(√f) for some non-zero f ∈ k(C). Then

P ∈ C is ramified ⇐⇒ ordπ(P )(f) is odd.

Proof. For notational convenience we write k(C) as an extension of k(C) since π∗k(C) ⊆ k(C).

Let P ∈ C(k) be a point and P := π(P ). Since C is smooth OC,P

is the integral closure of OC,Pin k(C). (See [8, Problem 7.20].)

First we assume that ordC,P

f is either 0 or 1. Notice

p(T ) := T 2 − f

is the minimal polynomial for√f overOC,P . Let S be the freeOC,P module generated by

1,√f

and note that S ⊆ OC,P

. By [7, I.4 Proposition 6ii] and [7, I.3 Proposition 4i] we have

Nmk(C)/k(C)

(p′(√f))OC,P = (4f)OC,P ⊆ Disc(S/OC,P )


where Disc(C/OC,P ) is the discriminant of S over OC,P (See [7, I.3 Equation 4]). If ordC,P f = 0

then 4f is a unit and the discriminant contains OC,P . Thus Disc(S/OC,P

) = OC,P and by [7, I.5

Theorem 1] the extension k(C)/k(C), as discrete valuation fields with discrete valuations ordC,P

and ordC,P (respectively), is unramified. That is, for a uniformizer t ∈ OC,P we have

ordC,P

t = 1.

Therefore P is an unramified point of π. Otherwise, if ordC,P f = 1 then f is a uniformizer for

OC,P . We see that

2 · ordC,P

√f = ord

C,Pf ≥ 1.

Since ordC

√f is an integer we have that ord

C,Pf > 1. Thus P is a ramified point of π.

We now address the general case. Let t be a uniformizer for OC,P and write f = ut2m+r where

u ∈ O∗C,P , m ∈ Z, and r ∈ 0, 1. Then since f · t−2m ∈ OC,P and

(√f · t−m

)2− f · t−2m = 0

we have that√f · t−m is integral overOC,P . Hence it is inO

C,P. We replace f with f · t−2m in the

previous argument to complete the proof.

Hyperelliptic curves

Definition 2.4.40. A curve C is called hyperelliptic if g(C) > 0 and there is a degree 2 map

π : C → P1.

Remark 2.4.41. If g(C) > 1 then π is determined up to an automorphism of P1. This is quite inter-

esting but we do not require this result. The alert reader will notice that we refer to "the hyperelliptic

involution" instead of "a hyperelliptic involution".

Remark 2.4.42. Recall the characteristic of k is not 2. Thus for any hyperelliptic curve C defined

over k we can find a squarefree f ∈ k[x] such that

X := V (y2 − f) ⊆ A2k

is an affine model of C.


Any separable double cover of curves is automatically Galois. In particular a hyperelliptic curve

C is a double cover of the projective line, so there is an automorphism of C corresponding to

changing the branches of this cover.

Definition 2.4.43. Let C be a hyperelliptic curve double covering P1. The involution of C over P1

is called the hyperelliptic involution.

Hyperelliptic curves are interesting because they are very easy to construct and it is very easy to

find the ramification locus of the map π : C → P1, as demonstrated by the corollary below.

Corollary 2.4.44. The ramification index of a point P ∈ H with respect to the quotient by hyperel-

liptic involution on a hyperelliptic curve is 2 if P is invariant under the hyperelliptic involution and

is 1 otherwise.

Proof. Since the degree of the quotient map is 2 that means the ramification index of each point is

either 1 or 2. By Proposition 2.4.8 we see that the result is immediate.

2.5 Abelian varieties

In this section we make precise what types of objects we are classifying and describe Jacobian

varieties. For a deeper look into the theory of abelian varieties the reader is encouraged to refer to

[13] or [14].

2.5.1 Definition and properties of abelian varieties

Definition 2.5.1. Let A be a smooth projective variety defined over k and O some distinguished

point over k on A. Furthermore suppose there are morphisms

+: A×A → A

[−1] : A → A

satisfying the usual associativity, inverse, and identity conditions. Then we call the quadruple

(A,O,+, [−1]) an abelian variety. We will refer to this data by A when the group structure is

clear from context.

We should point out that [14] begins with a different definition and then shows that the definition

given here is equivalent.


Theorem 2.5.2. The triple (A(k),+, O) defines a commutative group.

Proof. See [14, Corollary I.1.4].

Definition 2.5.3. We say that a morphism φ : A→ B of varieties is a morphism of abelian varieties

(A,+A, OA,−1A)→ (B,+B, OB,−1B) provided that φ(OA) = OB and φ(P +AQ) = φ(P ) +B

φ(Q) for all P,Q ∈ A.

Remark 2.5.4. The fibre over OB characterizes the fibre structure of the map φ. For any point

P ∈ B choose a Q such that φ(Q) = P . Then by additivity of φ we have that

φ−1(P ) =Q′ ∈ A : Q′ +A [−1]AQ ∈ φ−1(OB)

.

Since φ is a morphism on the level of groups we call φ−1(OB) the kernel.

We highlight a particularly useful family of morphisms.

Definition 2.5.5. The multiplication by m morphism, denoted [m], is defined by

[m]P := P + . . .+ P︸︷︷︸m times

.

Its kernel is called the m-torsion of A and is denoted A[m].

Remark 2.5.6. Since [m] is a morphism, OA is Zariski-closed, and the pullback of a Zariski-

closed set by a morphism is also Zariski-closed, we see A[m] can be given the structure of an

algebraic set. By [14, Theorem I.7.2] #A[m] is finite.

We shall now proceed to define an important type of morphism of abelian varieties and show

that this gives rise to a type of invariant known as the isogeny class. We then sharpen this informally

so that we may frame our motivating classification question in the correct language.

Definition 2.5.7. An isogeny of abelian varieties φ : A → B is a surjective morphism of abelian

varieties with finite kernel. If an isogeny exists we say that A is isogenous to B, denoted A ∼ B.

Lemma 2.5.8. Let U ⊆ A be a non-empty Zariski-open set. Then the collection of translates of U

⋃P∈U

UP := Q ∈ A : Q− P ∈ U


is an open cover of A.

Proof. It suffices to show that there is a translate U ′ of U containing the identity since 0 ∈ U ′

implies Q ∈ U ′Q. Observe that since [−1] is an automorphism we have

U ∩ [−1]U

is an open set. Since A is a variety, the intersection of any two non-empty open sets is again non-

empty so we may pick a point P ∈ U ∩ [−1]U . Immediately 0 ∈ UP and the rest of the result

follows.

Lemma 2.5.9. A ∼ B is an equivalence relation.

Proof. See [14, Remark 8.6].

Lemma 2.5.10. Let φ : A→ B and τ : A→ C be isogenies defined over k such that

τ∗(k(C)) ⊆ φ∗(k(B)) ⊆ k(A). Then there exists an isogeny ψ : B → C defined over k such that

τ = ψ φ.

Proof. Since φ and τ are isogenies they are surjective (and hence dominant). So φ∗(k(B)) ∼= k(B)

and τ∗(k(C)) ∼= k(C). Thus there exists a rational map ψ : B → C such that τ = ψ φ on the

open set for which ψ is defined. In particular, this means that whenever P,Q, P +Q ∈ U we have

ψ(P +Q) = ψ(φ(P ′) + φ(Q′)) = τ(P ′ +Q′) = ψ(φ(P ′)) + ψ(φ(Q′)) = ψ(P ) + ψ(Q).

On each open set UP we define ψP by

ψP (Q) := ψ(Q− P ) + ψ(P )

and define

ψ := (UP , ψP )P∈U .

To clarify the above definition, we mean for each UP and each Q ∈ UP that

ψ(Q) := ψP (Q).

We see this is well defined since ψ(Q) is independent of the choice of open set containing Q. By

Lemma 2.5.8 the open sets are a cover of C. It is not much more work to show that ψ is a legitimate


morphism from B to C such that ψ∣∣∣U

= ψ. Finally,

ψ(0B) = ψ(0B) + ψ(0B)⇒ ψ(0B) = 0C

and ψ is surjective since τ = ψ φ is. Thus ψ : B → C is an isogeny.

Definition 2.5.11. The isogeny class of A is the (∼)-equivalence class of A.

Next we introduce the product abelian variety. We will need a couple of definitions before proceed-

ing.

Proposition 2.5.12. Let X,Y be projective varieties defined over k. Then there is a projective

variety Z unique up to isomorphism such that

(a) There are morphisms π1, π2 defined over k such that π1 : Z → X and π2 : Z → Y are surjec-

tive.

(b) For any projective variety V admitting morphisms φ1, φ2 into X and Y (respectively) there

exists a unique morphism φ : V → Z such that the following diagram commutes:

X

Vφ //

φ1

77

φ2

''

Z

π1

>>

π2

Y

Proof. See [19, I.5.1] or [8, 6.4.6].

Definition 2.5.13. The Z produced by the above theorem is called the product variety of X and Y

and we write Z = X ×k Y .

Proposition 2.5.14. Let Z = X ×k Y . If P ∈ X(k) and Q ∈ Y (k) are smooth points then

(P,Q) ∈ Z(k) is a smooth point as well.

Proof. It suffices to check smoothness locally so we choose affine open sets Ux, Uy containing P,Q

respectively and notice that Ux × Uy is an affine open subset of Z containing P,Q. Then

OZ(Ux ×k Uy) ∼= OX(Ux)⊗k OY (Uy).


(See [19, Examples I.2.1.5, I.2.2.4]). By OX(Ux) we mean the affine co-ordinate ring of the affine

variety Ux. We then verify that

OZ,(P,Q)(Ux ×k Uy) ∼= OX,P (Ux)⊗k OY,Q(Uy).

and that for the associated maximal ideals

dimkM(P,Q)/M2(P,Q) = dimkMP /M

2P + dimkMQ/M

2Q = dimX + dimY = dimZ

so smoothness is verified.

Proposition 2.5.15. Let (A,OA,+A, [−1]A), (B,OB,+B, [−1]B) be abelian varieties. Let

C = A×k B

OC = (OA, OB) ∈ C

+C = (+A,+B)

[−1]C = ([−1]A, [−1]B).

Then (C,OC ,+C , [−1]C) is an abelian variety defined over k.

Proof. We remark that since A and B are smooth projective varieties defined over k then so is C.

We need to show that

• +C is a morphism on the level of varieties.

Since C is the product object, there is a map +C such that the following diagram commutes:

A×A +A // A

C × C

πA×πA99

+C //

πB×πB %%

C

πA

??

πB

B ×B +B // B

• [−1]C is a morphism.


As before we see commutativity of the diagram

A[−1]A // A

C

πA

>>

[−1]C //

πB

C

πA

>>

πB

B

[−1]B // B

gives us the result.

• OC is the identity.

First we show that OC +C P = P +C OC = P . By definition we have πA(OC) = OA and

πB(OC) = OB . Since

πA(P ) +OA = πA(P )

πB(P ) +OB = πB(P )

and C = A×k B, we must have that

P +OC = P.

A similar proof works for the other equality.

• +C is associative.

Again we have for P,Q,R ∈ C that

πA(P ) + (πA(Q) + πA(R)) = (πA(P ) + πA(Q)) + πA(R)

πB(P ) + (πB(Q) + πB(R)) = (πB(P ) + πB(Q)) + πB(R).

Therefore P + (Q+R) = (P +Q) +R.

• [−1]C is inverse.

Same trick.


Definition 2.5.16. A k-isogeny factor of an abelian variety C defined over k is a non-zero abelian

variety A defined over k such that there exists an abelian variety B defined over k such that there is

an isogeny φ : A×B → C defined over k.

2.5.2 Definition and properties of the Jacobian

In this section we will define the Jacobian variety of a curve as the abelian variety with the same

group structure as the Picard group of the curve and discuss how maps of curves give rise to induced

morphisms of their Jacobians. Again the interested reader is directed to [14, Section III].

Theorem 2.5.17. Let C be a smooth projective curve. Then there is an abelian variety called the

Jacobian of C such that in a natural way:

J(C)(k) ∼= Pic0(C).

By natural we mean that given a surjective morphism of curves π : C → C we have π∗ : J(C) →J(C) and π∗ : J(C)→ J(C) are morphisms as abelian varieties.

Proof. See [14, Theorem III.1.2, Remark III.1.4a].

Theorem 2.5.18. If C is a smooth projective curve then dim J(C) = g(C).

Proof. See [14, Proposition III.2.1].

2.5.3 Polarizations, principal polarizations, and polarized isogenies

The purpose of this section is to emphasize that the decompositions of Jacobian varieties as prin-

cipally polarized abelian varieties are indeed quite stringent and worth pointing out whenever they

occur. We only need formal properties of polarizations so the definitions we state here are incom-

plete. A proper treatment of polarizations and the definition of the dual abelian variety is beyond the

scope of this thesis but can be found in [13] or [14].

Proposition 2.5.19. If A is an abelian variety then there is a dual abelian variety denoted A∨. We

also call A∨ the Picard Variety of A and denote it by Pic(A).

Proposition 2.5.20. Let A be an abelian variety and A∨ its dual. Then

dimA = dimA∨.


Proof. See [14, Remark I.8.7e].

Proposition 2.5.21. Let A,B be abelian varieties. Then (A×B)∨ ∼= A∨ ×B∨.

Proof. See [13, Proposition IV.4.7].

Definition 2.5.22. A polarization is a special type of isogeny λ : A → A∨. A polarization is said

to be principal if # kerλ = 1. A pair consisting of an abelian variety and a specified (principal)

polarization is called a (principally) polarized abelian variety.

Proposition 2.5.23. If λ ∈ Hom(A,A∨) is a polarization and n ∈ Z is nonzero then nλ 6= 0.

Proof. See [14, Lemma I.10.6] or [14, Lemma I.10.18].

Proposition 2.5.24. If φ : A→ B is an isogeny of abelian varieties then there is an induced isogeny

φ∨ : B∨ → A∨ of the same degree.

Proof. See [14, Theorem I.9.1].

Definition 2.5.25. Let (A, λA), (B, λB) be polarized abelian varieties. Then an isogeny φ : A→ B

is said to respect polarizations if there are non-zero n,m ∈ Z such that the diagram

A

φ

nλA // A∨

BmλB // B∨

φ∨

OO

commutes. As it turns out nm = ±deg λB ·(deg φ)2

deg λA. We say φ is a polarized isogeny.

Proposition 2.5.26. The map [n] : A→ A respects polarizations.

Proof. Direct from definitions and the fact that [n]∨A = [n]A∨ .

Polarized abelian varieties, together with morphisms of abelian varieties respecting polariza-

tions, define a category. We discuss some of the properties of this category in that there are products

and the universal property of quotients.

Proposition 2.5.27. Let (A, λA), (B, λB) be polarized abelian varieties. Then (A × B, λA ⊕ λB)

is the product object in the category of polarized abelian varieties.


Proof. This is immediate from the fact that A × B is the product object in the category of abelian

varieties and the choice of polarization on A×B.

Proposition 2.5.28. Let α : (A, λA) → (B, λB) and β : (A, λA) → (C, λC) be isogenies of prin-

cipally polarized abelian varieties such that β∗(k(C)) ⊆ α∗(k(B)). Let γ : B → C be the unique

morphism such that β = γ α. Then γ respects polarizations.

Proof. Let nα = degα and nβ = deg β = deg γ · degα. Since degα divides deg β there is an

m ∈ Z such that nβ = mnα =: n. Since α and β respect polarizations and β∨ = α∨ γ∨ we have

that

nλA = α∨ mλB α

nλA = β∨ λC β

= α∨γ∨ λC γα.

So α∨(γ∨ λC γ−mλB)α = 0. Since α is surjective we see that α∨(γ∨ λC γ−mλB) = 0

and hence Im(γ∨ λC γ − mλB) ⊆ kerα∨. But α∨ is an isogeny of degree nα (Proposition

2.5.24) so

0 = [nα](γ∨ λC γ −mλB).

It follows that

BnαmλB//

γ

B∨

CnαλC // C∨

γ∨

OO

commutes.

We end this section by noting Jacobian varieties can be considered as polarized abelian varieties

and state the some important results regarding polarizations on Jacobian varieties.

Proposition 2.5.29. The Jacobian variety of a curve C admits a canonical principal polarization

coming from C, denoted by λC .

Proposition 2.5.30. Let π : C → C be a morphism of curves and π∗ : J(C) → J(C) the induced


map on the Jacobians. Let λC, λC be the canonical polarizations on J(C), J(C) respectively. Then

J(C)λC //

π∗

J(C)∨

(π∗)∨

J(C) J(C)∨

λ−1Coo

commutes.

Proof. See [15, Section 1].

Proposition 2.5.31. LetC be a curve defined over k and letP ∈ C(k). Then there exists a morphism

jP : C → J(C) defined by

jP (Q) := [Q− P ]

where [Q−P ] is the point on J(C) corresponding to the element [Q−P ] ∈ Pic0(C). (See Theorem

2.5.17.)

Proof. See [10, Theorem A.8.1.1].

Theorem 2.5.32 (Torelli). Let C and C ′ be smooth projective curves over an algebraically closed

field k, and let jP : C → J and jP ′ : C ′ → J ′ be the maps of C and C ′ into their Jacobians

defined by points P and P ′ on C and C ′. Let β : (J, λC) → (J ′, λC′) be an isomorphism from the

canonically polarized Jacobian of C to that of C ′.

(a) There exists an isomorphism α : C → C ′ such that jP ′ α = ±β jP + c for some c in J ′(k).

(b) Assume that C has genus ≥ 2. If C is not hyperelliptic, then the map α, the sign ±; and c are

uniquely determined by β, P, P ′. If C is hyperelliptic, the sign can be chosen arbitrarily, and

then α and c are uniquely determined.

Proof. See [14, Theorem III.12.1].

2.5.4 Decompositions of the Jacobian

Up until now we have merely treated the Jacobian variety as an abstract group and mentioned that

the group aspects we had talked about correspond to geometric operations. We now discuss decom-

positions of Jacobian varieties as abelian varieties.


Definition 2.5.33. Let A,B and C be nontrivial principally polarized abelian varieties. We say that

C decomposes as polarized abelian varieties into A and B if there exists a polarized isogeny φ such

that

φ : A×B → C.

We highlight the particular type of decomposition we are interested in.

Definition 2.5.34. Let φ : A× B → C be a decomposition as polarized abelian varieties of C into

non-trivial principally polarized abelian varieties. Suppose that ψ : A[n]→ B[n] is an isomorphism

both as abstract groups and as algebraic sets. If

kerφ = (a,−ψ(a)) ∈ A[n]×B[n] : a ∈ A[n]

we say that C is the principally polarized abelian variety obtained by gluing A and B along their

n-torsion.

2.6 Endomorphisms of abelian varieties

Let A be an abelian variety.

Definition 2.6.1. An endomorphism of A is a morphism of abelian varieties φ : A → A such that

φ(0A) = 0A and φ(x+ y) = φ(x) + φ(y).

• The identity morphism 1 is an endomorphism. It is defined over k.

• The trivial morphism 0 defined by 0(x) = 0A is also a morphism defined over k.

Proposition 2.6.2. If φ, ψ are endomorphisms of A then φ+ ψ, φ ψ are also an endomorphisms.

Proof. The ring criteria are straightforward to check and the composition of morphisms of abelian

varieties is also a morphism of abelian varieties. Thus we conclude φ ψ is an endomorphism of

abelian varieties. All that is left to assert is that φ+ ψ is a morphism as varieties. But we see by the

diagram

Adiag // A×A φ⊕ψ // A×A + // A

that φ + ψ is a composition of morphisms of varieties. On the level of groups we see that for

P,Q ∈ A(φ+ ψ)(OA) = φ(OA) + ψ(OA) = OA


(φ+ ψ)(P +Q) = φ(P ) + φ(Q) + ψ(P ) + ψ(Q) = (φ+ ψ)(P ) + (φ+ ψ)(Q).

Proposition 2.6.3. There exists an endomorphism [−1] which satisfies the inverse properties that

one would expect. Namely for any endomorphism φwe have [−1]φ = φ[−1] and φ+[−1]φ = 0.

Proof. Since A is an abelian variety there is an inverse morphism [−1]A. Let P ∈ A. Then

(φ+ [−1]Aφ)(P ) = φ(P ) + [−1]Aφ(P ) = 0

φ([−1]AP ) + φ(P ) = φ(P + [−1]AP ) = φ(0) = 0.

These lead to the natural definition:

Definition 2.6.4. The endomorphism ring of an abelian variety End(A) is the ring with ring struc-

ture (0,1,+, ) specified above.

End(A) gives us a lot of useful information about A. Since abelian varieties are projective we

get the following lemma:

Lemma 2.6.5. Let φ ∈ End(A). Then φ(A) is a sub-abelian variety of A.

Lemma 2.6.6. Let C, C be curves, π : C → C, and σ ∈ Aut(C/C). Then the action of σ∗ on

Div0(C) induces an endomorphism of J(C) by

σ∗([D]) = [σ∗(D)].

Moreover, π∗ σ∗ = π∗ and σ∗ π∗ = π∗.

Proof. First we have to show that σ∗(Princ(C)) ⊆ Princ(C). Let

div(f) := D =∑P∈C

aPP

be a principal divisor. Then

σ∗(D) =∑P∈C

aPσ(P ).


We see σ∗(D) is exactly div(f σ−1), which is a well defined function of C. Thus σ∗ acts compati-

bly on divisor classes. We also infer that σ∗([0]) = [0] and that σ∗([D]+[D′]) = σ∗([D])+σ∗([D′]).

Since σ : C → C by Theorem 2.5.17 we assert that σ∗ is a morphism on the level of varieties. Since

π σ = π, we have that

π∗ σ∗

∑P∈C

aPP

=∑P∈C

aPπ(σP ) = π∗

∑P∈C

aPP

.

We also see that

σ∗ π∗(∑P∈C

aPP

)= σ∗

∑P∈C

∑π(Q)=P

eQnPQ

=∑P∈C

∑π(Q)=P

eQnPσ(Q).

But π σ = π, so points in the fibre over P go to points in the fibre over P . Thus σ∗ π∗ = π∗.

Remark 2.6.7. The morphism in End(J(C)) induced by σ is denoted by σ∗.

Definition 2.6.8. Let R be a (not necessarily commutative) ring. An idempotent of R is an element

ε ∈ R such that ε2 = ε.

Endomorphism rings give us all the information we need to determine the isogeny factors of an

abelian variety. This is due to the classical result of Kani and Rosen [12], which we state with the

aid of the following lemma.

Lemma 2.6.9. End(A) is torsion-free. Equivalently, the map End(A) → End(A) ⊗Z Q given by

φ→ φ⊗ 1 is an injection.

Proof. See [14, Lemma I.10.6].

Theorem 2.6.10 (Kani-Rosen). Let A be an abelian variety. Let ε1, . . . , εn ∈ End(A) ⊗Z Q be

idempotents. Then idempotent relations correspond to isogeny relations between abelian varieties.

In particular,

(a) If ε ∈ End(A) ⊗Z Q is an idempotent then we may find an m ∈ Z such that m · ε ∈ End(A).

Moreover mε(A) is also an abelian variety.


(b) if∑

i εi = 1 then there is an integer m such that

A ∼ mε1A× . . .×mεnA

and conversely, if A ∼ B1 × . . .×Bn then we may find idempotents ε1, . . . , εn and integers mi

such that

miεi(A) ∼ Bi

and

A ∼ m1ε1A× . . .×mnεnA.

2.7 Final preliminaries

This last section covers some technical lemmas and contextual results which we isolate here in order

to improve readability of the next chapter.

2.7.1 Motivating facts for the case g = 2

The following results classify all principally polarized abelian varieties of dimension 2. This greatly

simplifies the types of decompositions that we need to consider since we are only looking for Jaco-

bian factors.

Proposition 2.7.1. Every genus 2 curve is hyperelliptic.

Proof. Observe that the Riemann-Roch space of the canonical divisor has dimension 2. Choosing a

basis 〈f, g〉 we see that the map

(f, g) : C → P1

P → (f(P ) : g(P ))

is surjective and extends to a morphism on all of C. Both f and g are degree at most 2 since this

is the degree of the canonical divisor so the map has degree at most 2. The map has degree greater

than 1 since g(C) > 0.

Theorem 2.7.2. Every principally polarized 2-dimensional abelian variety is either the Jacobian

variety of some hyperelliptic curve C or is a product of elliptic curves E1 × E2.

Proof. See [21, Satz 2].


2.7.2 Representing 2-torsion points on hyperelliptic Jacobians

In this section we shall provide a concrete specification of the 2-torsion of the Jacobian of a hyper-

elliptic curve. We will represent these 2-torsion classes by divisors supported on special points of

the curve that are easy to identify.

Definition 2.7.3. A Weierstrass point on a genus 2 curve is a point P such that

`(2P ) > 1.

This is a bit of an awkward definition for our purposes, so we provide a practical criterion

Theorem 2.7.4. Let π : C → P1x be a hyperelliptic curve with hyperelliptic involution ι. Then

P ∈ C is a Weierstrass point if and only if eP > 1.

Proof. Let t be a uniformizer for π(P ) ∈ P1x. The reverse direction is easy since 〈1, 1

t 〉 ⊆ L(2P ).

For the forward direction let P ∈ C such that P 6= ι(P ). Then

L(2P ),L(2ι(P )) ⊆ L(2(P + ι(P ))).

So by the Riemann-Roch theorem

`(2(P + ι(P ))) = 3.

Clearly L(2(P + ι(P ))) = 〈1, 1t ,

1t2〉. Any k-linear combination of these functions has equal valua-

tions at P and ι(P ) so

L(2P ) = L(2P ) ∩ L(2P + 2ι(P )) = 〈1〉.

Theorem 2.7.5 (Hilbert 90). Let L/K be a finite cyclic extension of fields with Gal(L/K) = 〈σ〉and let f ∈ L. Then NmL/K(f) = 1 if and only if there is a g ∈ L such that f = g

gσ .

Proof. See [5, 14.2 Exercise 23].

Lemma 2.7.6. Let π : C → C be a double cover of curves with Aut(C/C) = 〈σ〉. Then for any

divisor class [D] ∈ J(C) with σ∗[D] = [D] we may find a divisor D′ of C (not necessarily defined

over k) such that σ∗D′ = D′ and [D] = [D′].


Proof. Let D be a representative for [D]. Since σ∗[D] = [D] we have that σ∗D − D = div f for

some f ∈ k(C). Then

div fσ + div f = 0

and in particular f ·fσ is a constant which we may assume to be 1. Since the Galois group is a finite

cyclic extension and the norm of f is 1 we may apply Hilbert 90 to find g ∈ k(C) such that

f =g

gσ.

NowσD −D = div(g)− div(gσ)

⇒ σD + div(gσ) = D + div(g).

Taking D′ = D + div(g) completes the proof.

Lemma 2.7.7. Let C be a hyperelliptic curve and [D] ∈ Pic0(C)[2]. Then we can find a represen-

tative D ∈ Div0(C) such that D is supported only on the Weierstrass points of C.

Proof. Let σ be the hyperelliptic involution and observe that a 2-torsion class must satisfy σ([D]) =

−[D] = [D]. Moreover the cover π : C → P1 is finite and cyclic, so by the previous lemma we can

find a divisor D′ such that

D′ =∑

θi Weierstrass points

aiθi + π∗(a)

where a ∈ Div(P1). Since π∗(a) ∼ deg(a) · θi we are done.

Chapter 3

Curves of genus 2g with decomposableJacobians

3.1 Introduction

In this chapter we shall make use of the terminology and machinery referenced in the previous

chapter and prove the main result of this thesis.

Definition 3.1.1. Let G and H be finite abelian groups and let ψ : G→ H be an isomorphism. We

call the subgroup

∆ :=

(g, h) ∈ G×H : h = ψ(g)−1

the anti-diagonal of G×H .

Definition 3.1.2. Let S2 be the symmetric group on 2 elements. Let V be a variety, letM be a set

of varieties, and let P2 be the set of pairs in P1k(k)× P1

k(k)/S2 such that

(i) P1 6= P2

(ii) either P1, P2 are both k-rational points or P1 is the quadratic conjugate of P2.

Then a two parameter family (associated to V ) is the image of a map of sets ϕ : V × P2 →M.

Remark 3.1.3. The definition we use for a two-parameter family is sufficient to state the main

result but lacks the requirements for ϕ to be continuous and for independence of the parameters. It

is beyond the scope of this thesis to provide a full treatment of parameter families.

35

CHAPTER 3. CURVES OF GENUS 2G WITH DECOMPOSABLE JACOBIANS 36


genus g curve defined over k, and J(Cf ) its Jacobian. Then we may find a two parameter family of



ψ : J(CF )[2]→ J(Cf )[2].


3. J(A) ∼= J(Cf )× J(CF )/∆ as polarized abelian varieties, where ∆ is the (anti)-diagonal of

J(Cf )[2]× J(CF )[2].

Proposition 3.1.4. Let A be a genus 2g double cover of a hyperelliptic genus g curve CF , which

double covers P1x. Let Ω be the Galois closure ofA/P1

x and assume thatA 6= Ω. Then Gal(Ω/P1x) ∼=

D4. Moreover, there is a choice of Cf and parameters as in the above theorem where we recover A

and CF .

It is useful to know when this occurs for a number of reasons since J(A) decomposing in this

way may allow us to say something interesting about either A or one of the component Jacobians.

We list some potential applications:

• The endomorphism ring of J(A) can inherit special properties of the endomorphism rings of

J(CF ) and J(Cf ).

• We can show any principally polarized abelian of dimension 2 arises as an isogeny factor (de-

fined over k) of a Jacobian of a genus 4 curve. More generally, we can show any hyperelliptic

g-dimensional Jacobian arises as an isogeny factor defined over k of some Jacobian of a genus

2g-curve.

The proof of the main result will proceed as follows. First we shall review the historical literature

both to show the inspiration for the main construction and to show potential applications for it.

We then provide the main construction for the curves A and CF and calculate some necessary

information. We will prove the main result and finally list some of its corollaries and potential

future directions.


3.2 Construction 1: Legendre

The identification of Jacobian varieties that are gluings of smaller Jacobian varieties has seen a

number of historical uses. See for example [1, 3]. One hopes with a generalized construction the

techniques already present in the literature can be considerably extended. Out of historic respect and

conceptual insight we review the classical construction to show the origin of the method employed

by this thesis.

For f ∈ k[x] a square-free quintic, a ∈ k such that f(a) 6= 0, and d ∈ k non-zero we let

C1 : y2 = f(x)

C2 : z2 = d(x− a)f(x).

Proposition 3.2.1. Let L = k(x)(√f,√d(x− a)). Then L/k(x) is Galois with Galois group V4.

Proof. Since d(x− a) is not a square multiple of f(x) we have

√d(x− a) 6∈ k(x)(

√f).

Thus [L : k(C1)] = [L : k(C2)] = 2 and [L : k(x)] = 4. Since every separable extension of

degree 2 is Galois, we see that L/k(C1) and L/k(C2) are both Galois. Hence L is Galois over

k(C1)∩k(C2) = k(x). Finally, since k(C1) 6= k(C2) we see that L does not have a unique subfield

of index 2. Thus Gal(L/k(x)) is not cyclic and so Gal(L/k(x)) ∼= V4.

Let Ω be the curve corresponding to the composite field of k(C1) and k(C2). By Proposition

3.2.1 we have the familiar diagram of Figure 3.1.

Ωα

~~

β

C1

π

V

C2

ρ~~P1

Figure 3.1: Subcover structure of Ω/P1.

Lemma 3.2.2. Let V and Ω be as in Figure 3.1. Then g(V ) = 0, g(Ω) = 4.


Proof. From function fields we see k(V ) = k(x)(yz) = k(x)(√d(x− a)f(x)2) is ramified at two

points, a and∞. Since this is a degree 2 extension we have that V ∼= P1k.

Notice that there is a single Weierstrass point P onC1 lying over∞ and that there are two points

on C2 lying over infinity. Thus there are at least two points on Ω lying over∞ since Ω is a cover of

C2. Thus α−1(P ) = β−1ρ−1(∞) contains at least two points, so no points over P are ramified.

Hence k(Ω) = k(C1)(√d(x− a)) is ramified at only two points. Thus from the Riemann-Hurwitz

formula g(Ω) = 4.

Lemma 3.2.3. Let Ω, C1, C2 be as in Figure 3.1. Then J(C1)[2] ∼= J(C2)[2]. Moreover, if ∆ is the

anti-diagonal of J(C1)[2]× J(C2)[2] then J(Ω) ∼= J(C1)× J(C2)/∆.

Proof. Let α : Ω → J(C1), β : Ω → J(C2), τ be the nontrivial automorphism of Ω/C1, and σ

be the non-trivial automorphism of Ω/C2. By Lemma 2.6.6 there are endomorphisms τ∗, σ∗ ∈End(J(Ω)). By Lemma 2.6.6 we see that α∗ τ = α∗. Hence Im(1−τ∗) ⊆ ker(α∗) and

1−τ∗ = 1−τ∗ + (1+τ∗ + σ∗ + σ∗τ∗)− (1+σ∗τ∗) = 1+σ∗ ∈ End(Ω).

We see ker(α∗) ⊆ Im(1+σ∗) = Im(β∗). Similarly ker(β∗) ⊆ Im(α∗). Therefore

J(C1)× J(C2)α∗+β∗

&&[2]

J(Ω)

(α∗,β∗)xxJ(C1)× J(C2)

commutes.

We now proceed to compute the kernel which must be contained in J(C1)[2]×J(C2)[2]. Notice

that f ∈ k(P1x) has a divisor of the form

div f = θ1 + . . .+ θ5 − 5∞

where the θi are the points corresponding to the roots of f(x) on P1x. Let D := θi + θj − θk − θl ∈

Div0(P1x) for arbitrary i, j, k, l ∈ 1, . . . , 5. Since all of these points lie under ramification points


of both π and ρ we see that the divisors

D1 :=1

2π∗(D) =

1

2

(2π−1(θi) + 2π−1(θj)− 2π−1(θk)− 2π−1(θl)

)D2 :=

1

2ρ∗(D) =

1

2

(2ρ−1(θi) + 2ρ−1(θj)− 2ρ−1(θk)− 2ρ−1(θl)

)are well defined. It is also clear that for the map γ : Ω→ P1

x that

α∗([D1]) + β∗([D2]) = γ∗([D]) = [0].

Thus it is immediate that ker(α∗ + β∗) is the anti-diagonally embedded 2-torsion since the D1, D2

generate J(C1)[2], J(C2)[2] respectively.

We intend to produce a generalization of this construction. One particular restriction of Legen-

dre’s construction is that we insist both C1 and C2 have a rational Weierstrass point (over ∞ and

over α respectively). Not every genus 2 curve has to have a rational Weierstrass point.

3.3 Diagrams associated to a Galois covering

The main technique we will use to construct Jacobians of genus 2g curves that arise as a gluing of

Jacobians of genus g curves is to construct Galois covers which have these objects as isogeny factors

and see if these can be recognized as isomorphisms. If A/P1x is not already Galois then the Galois

closure (Ω) of the tower of double covers A → H → P1x over P1

x will have Gal(Ω/P1) ∼= D4. We

give names to the automorphisms and the curves arising from Ω.

Lemma 3.3.1. Let K,L,M be fields and let M/K and L/M be Galois extensions of degree 2. If

L/K is not Galois then there is a degree 2 extension L/L such that L/K is Galois and Gal(L/K) ∼=D4.

Proof. Let L = M(α) and let p be the minimal polynomial of α over M . Let σ be the nontrivial

automorphism of M/K. Then σ acts on p by acting on the coefficients of p. Let β be a roots of pσ

and define L′ := M(β).

IfL′ = L this is a contradiction, since then both 〈σ〉 and Aut(L/M) are subgroups of Aut(L/K).

This would imply σ ∈ Aut(L/M) and so fixes M , a contradiction.


Define L := M(α, β). We see that [L : M(α)] = 2 and that L is the splitting field of the

polynomial p · pσ whose coefficients are in M . Thus L/M is Galois and there are two distinct

degree 2 sub-extensions of L/K, M(α) and M(β). Therefore Gal(L/M) ∼= V4. But p · pσ is σ

invariant and so has coefficients in K. We conclude L/K is Galois and furthermore:

(i)∣∣∣Gal(L/K)

∣∣∣ = 8,

(ii) Gal(L/K) has a V4 subgroup,

(iii) Gal(L/K) is non-abelian since L/K is not Galois.

Therefore Gal(L/K) ∼= D4.

3.3.1 Models

The following are affine models for the curves corresponding to Ω and its subcovers. By Theorem

2.3.9 we may find a smooth projective model in place of these objects and note that since the Jaco-

bian is determined by the function field that we can use the information provided by these models.

CF : V (y2 − F (x)) ⊆ A2k(x, y)

A1 : V (y2 − F (x), z21 − r + y) ⊆ A3

k(x, y, z1)

A2 : V (y2 − F (x), z22 − r − y) ⊆ A3

k(x, y, z2)

Ω : V (y2 − F (x), z21 − r + y, z2

2 − r − y) ⊆ A4k(x, y, z1, z2).

We remark that the given affine model of CF is a double cover of A1k by the map φ : (x, y) → (x).

Let P1x be the corresponding P1

k double covered by CF . If A1 is not Galois over P1k then, by Lemma

3.3.1, the curve Ω is Galois over P1k with Galois group isomorphic to D4. The automorphisms of Ω

over P1k are generated by

ι := (y, z1, z2)→ (−y, z2, z1)

ρ := (y, z1, z2)→ (y,−z1, z2)

τ := (y, z1, z2)→ (y, z1,−z2)

τι = (y, z1, z2)→ (−y, z2,−z1)

where y2 = F, z21 = r − y, z2

2 = r + y. Observe

(τι)2(y, z1, z2) = τι(−y, z2,−z1) = (y,−z1,−z2) = ρτ(y, z1, z2).


Since (τι)2 is an element of order 2, τι is of order 4. We also note that ρ and τ commute. We also

note that ιρ = τι.

〈1〉

ss !! ++〈ρ〉

""

〈τ〉

||

〈ρτ〉

vv ((

〈ι〉

""

〈ρτι〉

〈ρ, τ〉

((

〈ρι〉

〈ι, ρτ〉

vvD4

Figure 3.2: Subgroup structure of Aut(Ω/P1). Arrows denote inclusion.

We relate the subgroups in Figure 3.2 to their quotient curves (and corresponding fixed fields) in

Figure 3.3 through the usual correspondence of Galois theory.

Ω

tt ~~ **A2

A1

~~

V

ww ''

Y1

Y2

CF

((

V

X

wwP1x

Figure 3.3: Subcover structure of Ω.

We compute the genera of Y1 and X .

Lemma 3.3.2. Let Ω, CF , A1, Y1, X,P1x be as in Figure 3.3 and let g = g(CF ). Let π : A1 → CF ,

φ : CF → P1x, ξ : Y1 → X , and η : X → P1

x be the covering morphisms as in Figure 3.3. If π is

ramified at two points P1, P2 such that φ(π(P1)), φ(π(P2)) ∈ P1x are distinct points which are not

zeros or poles of F , then g(A1) = 2g, g(Y1) = g and g(X) = 0.


Proof. By the Riemann-Hurwitz formula we have

2g(A1)− 2 = 2(2g(CF )− 2) +∑P∈A1

(eP − 1) = 4g(CF )− 2.

Thus g(A1) = 2g. The function z1z2 ∈ k(Ω) is invariant under both ι and ρτ and not invariant

under τ . Thus z1z2 ∈ k(X) and z1z2 6∈ k(x). But

z1z2 =√

(r − y)(r + y) =√r2 − F .

By Corollary 2.4.39 we have that divCF (r − y) = nP1P1 + nP2P2 + 2D where both nP1 and nP2

are odd. Since r2 − F = Nmk(CF )/k(x)(r − y) we have

divP1x(r2 − F ) = nP1φ(P1) + nP2φ(P2) + 2D.

Since k(X) = k(x)(√r2 − F ) it follows from Corollary 2.4.39 that the cover η is ramified at two

points. By the Riemann-Hurwitz formula

2g(X)− 2 = 2(2g(P1x)− 2) + 2 = −2.

Thus g(X) = 0.

The functionw := z1+z2 ∈ k(Ω) is invariant under ι but not under ρτ . Thus k(Y1) = k(X)(w).

Observe that

w2 = 2(r + z1z2).

Let D = divX(r + z1z2). Since τ(z1z2) = −z1z2 we have

π∗(D) = divP1x

Nmk(X)/k(x)(r + z1z2) = divP1xF.

Thus the number of points of odd multiplicity inD is at least the number of points of odd multiplicity

in divP1xF . Since all of the ramification points of Ω → P1

x lie over points of odd multiplicity in

divP1xF , φ(π(P1)), or φ(π(P2)), the number of points of odd multiplicity in divX D is equal to

the number of points of odd multiplicity of divP1xF . Therefore by Corollary 2.4.39 the number of

ramification points of φ and ξ are equal. By the Riemann-Hurwitz formula we conclude g(Y1) =

g.


Lemma 3.3.3. Let Y1, X be as in Lemma 3.3.2. Then X has a rational point and is isomorphic to

P1k.

Proof. See [10, Exercise A.4.12b].

3.4 Construction of a dihedral cover of P1k

Lemma 3.4.1. For each genus g curve C separably double covered by a genus 2g curve A there

exists a function z ∈ k(A) such that k(A) = k(C)(√z) and the divisor of z is of the form

div(z) = 1 · P1 + 1 · P2 + 2(D+ −D−)

where D+, D− are effective and P1, P2 ∈ C(k).

Proof. Let z be a function such that k(A) = k(CF )(√z). By the Riemann-Hurwitz formula the

cover is ramified at two points so we write

div(z) = P1 + P2 + 2(D+ −D−)

with D+ and D− effective.

Remark 3.4.2. The divisor D− −D+ is both degree 1 and defined over k.

3.4.1 Norm construction

In this section we provide a construction that shows how, given hyperelliptic genus g curve Cf , to

obtain pairs (A,CF ) of curves defined over k such that A is a double cover of CF ramified at 2

points and CF is a hyperelliptic genus g curve. The primary utility of this lemma is the principle

given by Kani-Rosen (2.6.10) that a common Galois cover Ω of Cf , CF and A will give rise to

isogeny relations between J(Cf ), J(CF ) and J(A). We will proceed with a careful computation to

determine the exact isogeny relations.

Lemma 3.4.3. Let P1, P2 ∈ P1k

be points such that P1 6= P2. Assume either P1 and P2 are k-

rational points or P1 is the quadratic conjugate of P2. Then there is an involution µ : P1k→ P1

k

defined over k that fixes P1 and P2.


Proof. Fix a choice of co-ordinates (x : z) for P1k. We may assume up to translation that P1, P2 are

in the set (x : z) ∈ P1

k : z 6= 0

so we let P1 = (a : 1), P2 = (b : 1) for some a, b ∈ k. If P1 and P2 are rational then a+ b, ab ∈ k.

If P1 and P2 are quadratic conjugates then a and b are conjugate, that is

(t− a)(t− b) ∈ k[t].

So a+ b, ab ∈ k. We see the morphism µ : P1k→ P1

kdefined by

µ(x : z) := (x : z)

[a+ b 2

−2ab −a− b

]

is defined over k. A simple calculation shows that µ fixes P1 and P2 and is an involution.

Lemma 3.4.4. Let Rπ = P1 + P2 ∈ Div(P1k) be a degree 2 divisor such that P1 and P2 fulfil the

conditions of Lemma 3.4.3. Let µ be the involution fixing P1 and P2. Then we may find π : P1k → P1

k

such that

(i) there is an element x ∈ k(P1k) such that k[t, µ∗(t)] is an integral extension of k[x],

(ii) π∗(k(P1k)) = k(x), π has degree 2, andRπ is the ramification divisor of π.

Proof. If µ fixes∞ then µ∗(t) = c− t for some c ∈ k. Define the rational map

π : P1k → P1

k

(t : 1) → (tµ∗(t) : 1).

By Theorem 2.3.10 we see that π extends to a double cover. Let x := π∗(t) = t(c − t) and notice

that both t, µ∗(t) are roots of

p(T ) = T 2 − cT + x ∈ k[x][T ].


So (i) is satisfied in this case. If µ does not fix∞ then we define the rational map

π : P1k → P1

k

(t : 1) →(t+ µ∗(t)

2: 1

).

By Theorem 2.3.10 we see that π extends to a double cover. Let x := π∗(t) =(t+µ∗(t)

2

). If c ∈ k

we see thatdivP1

k(t− c) = (c : 1)−∞

divP1kµ∗(t) = µ(0)− µ(∞).

Since µ does not fix∞ we may choose c such that (c : 1) = µ(∞). If µ(0) 6=∞ then

divP1k(t− c)µ∗(t) = µ(0)−∞ = divP1

k((t− c) + α)

for some α ∈ k. Since t, µ∗(t) ∈ k(t) we can find d ∈ k such that

(t− c)µ∗(t) = d[(t− c) + α].

So (t− c)(µ∗(t)− d) = dα ∈ k. Both (t− c) and (µ∗(t)− d) are roots of

p(T ) = T 2 − xT − dα ∈ k[x][T ].

If µ(0) =∞ then tµ∗(t) ∈ k and both t, µ∗(t) are roots of

p(T ) = T 2 − xT − tµ∗(t) ∈ k[x][T ].

In each case we have satisfied (i).

Observe that k(x) is contained in the subfield fixed by µ∗. Hence since [k(t) : k(x)] = 2 we see

k(x) is the fixed field of µ∗. Since the quotient map π is ramified at its fixed points and µ has order

2, we are done.

Lemma 3.4.5 (Norm construction). Let Cf : w2 − f(t) be an affine model of a hyperelliptic genus

g curve where f is a squarefree polynomial in t. Let Rπ = P1 + P2 ∈ Div(P1k) be a degree 2

divisor such that P1 and P2 fulfil the conditions of Lemma 3.4.3. Let µ, π be as in Lemma 3.4.4.

Denote the preimage of π by P1t and denote the image of π by P1

x whose function fields are denoted


by k(t), k(x) respectively. If the points of odd degree multiplicity in divP1tf and the points of odd

degree multiplicity in divP1tfµ are disjoint then:

(i) The hyperelliptic curve defined by the affine model CF := V (y2 − 4 Nmk(t)/k(x)(f)) ⊆ A2k

has genus g.

(ii) Let β : CF → P1x be the cover extending the rational map β(x, y) = (x : 1). There is a double

cover α : A → CF such that A has genus 2g and the ramification divisor of α, denoted Rα,

satisfies

β∗α∗(Rα) = π∗(Rπ).

Proof. Let

r :=f + f µ

2, q :=

f − f µ2

.

In particular µ(r) = r, µ(q) = −q, and f = r+q. Since [k(t) : k(x)] = 2 and q is not fixed by µwe

see that k(t) = k(x)(q). Since q vanishes at bothP1 andP2 we write divP1t(q) = nP1P1+nP2P2+D

with D not supported at P1 or P2. Hence since divP1tq is invariant under µ we have that

divP1x

Nmk(t)/k(x)(q) = divP1x(−q2) = nP1π∗(P1) + nP2π∗(P2) + 2D′

= π∗(P1) + π∗(P2)− 2∞+ 2(D′ +∞).

We infer from Corollary 2.4.39 and the fact k(t) = k(x)(√q2(x)) that both nP1 and nP2 are odd.

Since the divisor class group of P1 is Z we rewrite this as

divP1x(h(x)) + 2 divP1

x(q(x))

where h, q ∈ k(x). In light of this we rewrite Nmk(t)/k(x)(q) = d · h(x)q(x)2 with d ∈ k. We may

assume that d = 1 by setting h(x) := d · h(x). Now define

F (x) := 4 Nmk(t)/k(x)(f) = 4 · f · (fµ) = 4r(x)2 − 4h(x)q(x)2.

Recall there are no points with odd multiplicity in both divP1tf and divP1

tfµ. In particular since

P1 and P2 are fixed by µ they have multiplicity zero in divP1tf . Moreover for each point P ∈ P1

t (k)

at most one of P, µ(P ) can occur in divP1tf with odd multiplicity. Since divP1

xF = π∗ divP1

tf we

see the number of squarefree roots of F and f is the same, so the hyperelliptic curve CF : y2−F (x)

also has genus g.


All that is left to do is to show that (ii) is satisfied. Let

w2 := 2r(x)− y.

Define A by the affine model

A =

(x, y, w) ∈ A3k : (x, y) ∈ CF and w2 = 2r(x)− y

and notice that k(A) = k(CF )(w). Proceeding we have

Nmk(CF )/k(x)(w2) = 4r(x)2 − F = 4h(x)q(x)2

so divP1x

Nmk(CF )/k(x)(w2) = π∗(Rπ) + 2D′. Since k(A) = k(CF )(w) we note that

α∗(Rα) = odd multiplicity terms of divCF (w2),

so this cover is ramified at 2 points. We now compute the genus g(A). By Riemann-Hurwitz

2g(A)− 2 = 2(2g(CF )− 2) + 2

and so g(A) = 2, g(CF ) = 2g.

Lemma 3.4.6. Let A,Cf , CF ,P1t ,P1

x, q, f, w be as in Lemma 3.4.5. Let Ω be the Galois closure of

Cf over P1x. Then Gal(Ω/P1

x) ∼= D4 and k(A) ⊆ k(Ω).

Proof. By Lemma 3.3.1 we have that Gal(Ω/P1x) ∼= D4. Let

p(T ) := (T 2 − f(t))(T 2 − fµ(t)) ∈ k(Ω)[T ].

We see that f(t) + fµ(t) = 2r(x) ∈ k(x),

p(T ) = T 4 − (f(t) + fµ(t))T 2 + F (x) ∈ k(x)[T ],

and p(T ) is irreducible over k(x). But p(√f(t)) = 0, so p(T ) splits in k(Ω). Letw1 =

√f(t), w2 =√

fµ(t) and observe these are roots of p(T ). We have that w1w2 =√f · fµ = y and so k(CF ) ⊆

k(Ω). Finally, let

p(T ) = T 2 − (2r(x)− y) ∈ k(Ω)[T ].


We see that

p(w1 − w2) = w21 + w2

2 − w1w2 − (2r(x)− y) = 0,

so any root of p lies in k(Ω). But by definition p(w) = 0, so it follows that k(A) ⊆ k(Ω).

It turns out that ramification divisors of covers give us slightly more information about the as-

sociated Jacobians than just their dimensions. We make a few useful observations here before pro-

ceeding.

Proposition 3.4.7. Let Ω, A, and Cf be as in Lemma 3.4.6. Then the covers Ω → Cf and Ω → A

are ramified.

Proof. First consider

Ω = Cf ×P1 Cfµ

xx ''

Cf

&&

V

Cfµ

wwP1t

and looking at the corresponding function fields we have the following diagram

k(Ω)

k(t)(√f)

γ∗77

k(t)(√f · fµ)

π∗

OO

k(t)(√fµ)

ξ∗gg

k(t)

gg OO 77

Since we chose µ such that there are no points of odd multiplicity in both divP1tf and divP1

tfµ

we have that divP1t(f · fµ) has 4g + 4 points with odd multiplicity. Hence V → P1

t is ramified at

4g + 4 points so g(V ) = 2g + 1. By symmetry the ramification divisors of γ and ξ have the same

degree. Now by Riemann-HurwitzκΩ = κ

V+Rπ

κΩ = κCf +Rγ

κΩ = κCfµ +Rξ.


Taking degrees and solving the resulting linear system we find that π is unramified and γ, η are each

ramified at 2g + 2 points. Let α : Ω → A be the double cover in Lemma 3.4.6. By the Riemann-

Hurwitz formula2g(Ω)− 2 = 2(2g(V )− 2)

(2g(Ω)− 2)− 2(2g(A)− 2) =∑P∈Ω

(eα,P − 1).

Since g(A) = 2g we conclude g(Ω) = 4g + 1 and α is ramified.

Corollary 3.4.8. For Ω and V as in Proposition 3.4.7 we have that g(Ω) = 4g + 1 and g(V ) =

2g + 1.

Remark 3.4.9. The technical conditions of Lemma 3.4.5 ensure that quotienting by µ creates a

squarefree polynomial and thus CF has exactly the ramification data needed to have genus g.

3.5 Verifying J(CF )× J(Cf)/∆ ∼= J(A)

3.5.1 Computing with endomorphisms of J(Ω)

The following helpful lemma appears in [15, section 3]. Since we only need the double cover version

we prove this directly.

Lemma 3.5.1 (Mumford). If π : C → C is a ramified cover of prime degree then π∗ : J(C) →J(C) is an embedding.

proof for degree 2. Observe that since π∗ π∗ = [2] that

kerπ∗ ⊆ J(C)[2].

Suppose for non-zero [D] ∈ J(C)[2] that π∗([D]) = [0]. Then by definition there is a function

f ∈ k(C) such that

div f = π∗(D)

and in particular

div Nm(f) = π∗π∗(D) = 2D

since D was a non-principal divisor f 6∈ k(C). Hence because [k(C) : k(C)] = 2 is prime we have

k(C) = k(C)(√f)


Thus the ramification divisor of π is given by

π∗(Rπ) = odd multiplicity terms in div Nm(√f)

of which there are none, so the ramification divisor is trivial.

We notice in our diagram that each of J(Cf ), J(D), J(CF ) must all embed into J(Ω). This

means that we can use endomorphisms of J(Ω) to construct maps explicitly between these objects.

We first show how to identify each of these in J(Ω) and then proceed with the main result. The

following lemma will aid in this process.

Lemma 3.5.2. Let π : C → C be a double cover with Galois group Gal(C/C) = 〈σ〉. Then

π∗J(C) = Im(1+σ) ⊆ J(C). Moreover if π is ramified the map π∗ : J(C) → Im(1+σ) is an

isomorphism.

Proof. The first equality is the definition of π∗ and the containment π∗(J(C)) ⊆ J(C) is also

straight from the definition. That the map is injective follows from Lemma 3.5.1. By definition

of π∗ the map is also an isogeny onto its image. By the proof of Lemma 2.5.9 we see π∗ factors

through multiplication by 1 so it is an isomorphism.

Lemma 3.5.3. Let π : C → C be a ramified double cover with Galois group Gal(C/C) = 〈σ〉.Then

1+σ = π∗ π∗ ∈ End(J(C)).

Proof. The first equality follows immediately from Lemma 2.4.20. That it is an endomorphism

follows from the fact it is a sum of endomorphisms.

By applying the above lemma twice we obtain

Corollary 3.5.4. Let C α // C ′β // C be ramified double covers with Gal(C/C) = 〈σ, τ〉 ∼=

C2 × C2. Then

(1+σ)(1+τ) = α∗β∗ β∗α∗ ∈ End(J(C)).

Proposition 3.5.5. Letπ : A→ CF

α : Ω→ A

γ : Ω→ Cf


be the covering maps as in Lemma 3.4.6 with corresponding maps on the Jacobians π∗, π∗, α∗, . . . , γ∗.

Then the following diagram commutes.

J(Cf )× J(CF )α∗γ∗+π∗

''[2]

J(A)

(γ∗α∗,π∗)wwJ(Cf )× J(CF )

Proof. By composing we get a map

J(Cf )× J(CF )(γ∗α∗(α∗γ∗+π∗),π∗(α∗γ∗+π∗)) // J(Cf )× J(CF ) .

Since for isogenies φ, ψ, ρ we have φ(ψ + ρ) = φψ + φρ it suffices to check that

(i) π∗π∗ = [2]

(ii) π∗α∗γ∗ = 0

(iii) γ∗α∗π∗ = 0

(iv) γ∗α∗α∗γ

∗ = [2].

(i) Given by Lemma 2.4.20.

(ii) Let [D] ∈ γ∗J(Cf ) = Im(1+ι). Then we may find a class [D′] ∈ J(Ω) such that [D] =

(1+ι)[D′]. Now by Corollary 3.5.4

α∗π∗ (π∗α∗)((1+ι)[D′]) = (1+ρ)(1+τ)(1+ι)[D′]

= (1+ρ+ τ + ι+ ρτ + ρι+ τι+ ρτι)[D′].

We recognize this as the pull back of a P1x divisor class and so it must be trivial. Since α∗π∗ is

an embedding we have (π∗α∗)[D] = 0.

(iii) Let [D] ∈ α∗π∗(J(CF )) = Im((1+ρ)(1+τ)). As before write [D] = (1+ρ)(1+τ)[D′].

Now as before

γ∗γ∗((1+ρ)(1+τ)[D′]) = (1+ι)(1+ρ)(1+τ)[D′] = 0.


Since γ∗ is an embedding we have γ∗([D]) = 0.

(iv) Notice thatγ∗α

∗α∗γ∗ − [2] = γ∗α

∗α∗γ∗ − γ∗γ∗

= γ∗(α∗α∗ − 1)γ∗

= γ∗((1+ρ)− 1)γ∗.

Let [D] = (1+ι)[D′] ∈ γ∗J(Cf ) = Im(1+ι). Now

ργ∗γ∗(ρ)(1+ι)[D′]

= ρ(1+ι)(ρ)(1+ι)[D′]

= (1+ρτι)(1+ι)[D′]

= (1+ι+ ρτ + ρτι)[D′].

Again this is the pullback of a P1t class and is trivial. Since ργ∗ is an embedding we conclude

γ∗α∗α∗γ

∗ − [2] = 0⇒ γ∗α∗α∗γ

∗ = [2].

Proposition 3.5.6. Let α, γ, π be as in Proposition 3.5.5. Then the isogeny α∗γ∗ + π∗ respects

polarizations.

Proof. By Proposition 2.5.30 we may replace (π∗)∨ by λCF π∗λ−1A and make similar substitutions

for α∗ and γ∗. It follows that

(α∗γ∗ + π∗)∨ = (λCf × λCF ) (γ∗α

∗, π∗) λ−1A .

Now by Proposition 3.5.5 we have that

(α∗γ∗ + π∗)∨λA(α∗γ

∗ + π∗) = (λCf × λCF ) (γ∗α∗, π∗) λ−1

A λA (α∗γ∗ + π∗)

= (λCf × λCF ) (γ∗α∗, π∗) (α∗γ

∗ + π∗)

= (λCf × λCf )[2].

Thus α∗γ∗ + π∗ respects polarizations.


3.5.2 Kernel data

In order to finish the main result we must identify the kernel of α∗γ∗ + π∗. Notice that the kernel

must be contained in J(Cf )[2]× J(CF )[2] so we restrict our attention there. First we show how to

explicitly write down representatives of 2-torsion classes on hyperelliptic curves.

Lemma 3.5.7. Let θi ∈ Cf be a Weierstrass point. Then there exists a point θi ∈ CF such that

π−1(θi

)= α(γ−1(θi)).

Proof. Since θi is a Weierstrass point ofCf , both θi and γ−1(θi), are fixed by ρτ . Moreover, θi ∈ P1t

is disjoint from the ramification locus of P1t → P1

x. Thus since ρ acts trivially on A we can push

down to get that α(γ−1(θi)) is fixed by τ ∈ Aut(A/CF ). But then α(γ−1(θi)) is a cover of π−1(Y )

for some Y ⊆ CF . Since π−1(Y ) cannot contain a ramification point of π we see

∣∣γ−1(θi)∣∣ = 2 ≥

∣∣π−1(Y )∣∣ > 1

and so the cover is degree 1. Moreover, since∣∣π−1(Y )

∣∣ = 2 and intersects trivially with the ramifi-

cation divisor we see that

Y =θi

.

Corollary 3.5.8. The mapψ : J(Cf )[2]→ (π∗J(CF )) [2]

[D]→ (π∗)−1α∗γ∗([D])

is an isomorphism.

Proof. It is immediate from Lemma 3.5.7 that

[α∗γ∗(θi − θj)] = [π∗(θi − θj)].

Since the [θi− θj ] generate J(CF )[2] by Lemma 2.7.7 the map we have written down is a surjection

of finite abelian groups and their finite underlying algebraic sets. The algebraic sets are of equal

cardinality so it is an isomorphism.


Proposition 3.5.9. Let ψ be the isomorphism above and let

∆ := ([D], [−1]ψ([D])) ∈ J(Cf )[2]× J(CF )[2] : D ∈ J(Cf )[2] .

Then ∆ = ker(α∗ γ∗ + π∗).

Proof. That ∆ is contained in the kernel is easy to show. For equality we let ([D′], [D]) ∈ J(Cf )[2]×J(CF )[2]. Then

(α∗γ∗ + π∗)([D′], [D]) = (α∗γ

∗ + π∗)([D′]− [D′], [D] + ψ[D′]) = (α∗γ∗ + π∗)(0, [D] + ψ[D′])

= π∗([D] + ψ[D′]).

Since π∗ is injective this is [0] if and only if [D] = [−1]ψ([D′]).

This in conjunction with Proposition 3.5.5 gives

Proposition 3.5.10. J(Cf )×J(CF )/∆ ∼= J(A). Which is to say that J(A) is obtained as a gluing

of hyperelliptic Jacobian varieties of dimension g along their 2-torsion.

3.6 Proof of the main result

In this section we give a proof of the main result.


genus g curve defined over k, and J(Cf ) its Jacobian. Then there exists a two parameter family of



ψ : J(CF )[2]→ J(Cf )[2].


3. J(A) ∼= J(Cf )× J(CF )/∆ as polarized abelian varieties, where ∆ is the (anti)-diagonally

embedded 2-torsion of J(Cf ).

Proof. The norm construction (Lemma 3.4.5) gives a two parameter family of (A,CF ) such thatCFis hyperelliptic and A is a double cover of CF , so (2) has been proven. Corollary 3.5.8 shows that

ψ : J(CF )[2] → J(Cf )[2] is an isomorphism, thus we have proven (1). Proposition 3.5.10 gives


that J(A) ∼= J(Cf )× J(CF )/∆ as abelian varieties and Proposition 3.5.6 shows the isomorphism

respects polarizations. This completes (3) and finishes the proof.

3.7 Corollaries

One immediate consequence of this construction is:

Corollary 3.7.1. Any Jacobian of a hyperelliptic genus g curve arises as an isogeny factor of some

Jacobian of a genus 2g curve where the isogeny φ is defined over k.

3.7.1 MAGMA script

We provide a MAGMA script to demonstrate how our construction can be used to create non-

hyperelliptic genus 4 curves with larger than expected automorphism groups. First we construct

a genus 2 curve Cf isogenous to a product of elliptic curves by using a technical lemma. Then with

a careful choice of involution µ we apply the norm construction (Lemma 3.4.5) to obtain a genus

2 curve CF which is also isogenous to a product of two elliptic curves. We apply the main result

to see that J(A) is isogenous to a product of four elliptic curves. We verify with MAGMA that the

automorphism group of A is larger than Z/2Z.

First we shall require two technical lemmas that ensure the correctness of the program. One

lemma allows us to generate genus 2 curves isogenous to a product of elliptic curves and the other

gives a family of µ such that the curve produced by the norm construction also has this property.

Lemma 3.7.2. Let a, b, c ∈ k∗ be distinct elements. Let C be the hyperelliptic genus 2 curve defined

by the affine model

C : y2 − (x2 − a)(x2 − b)(x2 − c).

We claim that J(C) is isogenous to a product of elliptic curves.

Proof. The quotient of C by the map η((x, y)) = (−x, y) gives the elliptic curve

E1 := y2 − (x− a)(x− b)(x− c).

Thus we obtain the idempotent relation in End(J(C))⊗Z Q

1 =

(1+η

2

)+

(1−η

2

).


By Theorem 2.6.10 we see J(C) ∼ E1 × A. Comparing dimensions we see dimA = 1 so A must

be an elliptic curve.

Lemma 3.7.3. Let C and η be as above. Let µ be an involution of P1t and let π : P1

t → P1x be the

quotient by µ. Assume that µη = ηµ and π∗(f) is a square-free polynomial of degree 6. Then the

genus 2 curve CF defined by the affine model

V (y2 − π∗(f))

is isogenous to a product of elliptic curves.

Proof. We need only show that η pushes down to an involution on CF , i.e, that it is an involution on

the roots of π∗(f). Write

π∗(f) = f · fµ = d26∏i=1

(t− ai)(t− µai).

Since µ and η commute

π∗(f)η = d26∏i=1

(t− η−1ai)(t− η−1µai)

= d26∏i=1

(t− η−1ai)(t− µ(η−1ai))

so η acts on the roots of π∗(f) ∈ k[x] as well. Thus (x, y)→ (ηx, y) is an automorphism of

CF := V (y2 − π∗(f)(x)).

Corollary 3.7.4. Let Cf and µ be as above. We apply the main result to obtain the isogeny relation

J(A) ∼= J(Cf )× J(CF )/∆ ∼ E1 × E2 × E′1 × E′2.

We now explain some of procedural details of the script. Given Cf and µ the script creates CFand A based on the explicit formulae given by the norm construction (Lemma 3.4.5). We see that A

must have an extra involution since CF has an extra involution.


Example

Our example computation was done over the finite field k := F101. We chose

f := (t2 − 1)(t2 − 4)(t2 − 9) and µ to be the involution µ(t) := 14t . Our computation gives

Cf = V (w2 − f)

CF = V (y2 − (65x6 + 2x4 + 11x2 + 78))

g(x) = 51x6 + 52x4 + 36x2 + 52

A = V (y2 + 36x6 + 99x4 + 90x2 + 23, 50x6 + 49x4 + 65x2 + 49 + 100y + z2) ⊆ A3k.

We assert g(CF ) = 2 in the program. The MAGMA command "AutomorphismGroup" assures

us that Aut(A) = V4.

Chapter 4

Future directions

We discuss some of the future directions of research we can pursue from this point. Specifically we

focus on the converse to the main theorem. We conjecture that the construction of the main theorem

is the only way the Jacobian of a non-hyperelliptic genus 4 curve A decomposes like

∆→ J(CF )× J(Cf )→ J(A)

where ∆ is the graph of the 2-torsion subgroup of J(Cf ). We provide a rough outline of the argu-

ment.

4.1 Converse to the main theorem

Suppose A is a genus 4 curve which is a double cover of a genus 2 curve C. Notice since C is

hyperelliptic it is a double cover of a P1. If A/P1 is Galois then it is hyperelliptic. Otherwise, the

Galois closure of A/P1 is dihedral and so A can be constructed from the norm construction. If we

can show that J(A) decomposing according to our restrictions implies that A is the double cover of

a genus 2 curve then the converse to the main result will follow. The ideal tool to investigate this

conjecture is the Torelli theorem. Throughout let

1. B1, B2 be principally polarized abelian varieties of dimension 2 such that there is an isomor-

phism ψ : B1[2]→ B2[2].

2. ∆ := (D,−ψ(D)) ⊆ B1[2]×B2[2] : D ∈ B1[2]

58

CHAPTER 4. FUTURE DIRECTIONS 59

Proposition 4.1.1. Let φ : B1 × B2 → J(A) be the morphism of principally polarized abelian

varieties as in the main result and let

τ ′ := 1B1 ⊕[−1]B2 : B1 ×B2 → B1 ×B2.

Then there is a non-trivial involution on J(A) respecting the polarization.

Proof. By Proposition 2.5.27 we see τ ′ respects polarizations. Moreover τ ′ fixes the kernel of φ.

Since char(k) 6= 2 and deg φ is a power of 2 (See [14, Theorem I.7.2]) we see that

k(B1×B2)/φ∗k(J(A)) is a separable extension of degree #∆. Let K/k be a finite extension such

that each P ∈ ∆ is a K-rational point and let L := K(B1 × B2). Then the map tP : B1 × B2 →B1 ×B2 given by

tP (x) := x+ P

is an automorphism (as varieties) of B1 × B2/J(A) defined over K. Thus each t∗P is an automor-

phism of L/φ∗K(J(A)), so the extension is Galois. Notice for any f ∈ τ ′∗φ∗K(J(A)) we have

that f is fixed by each t∗P , so f ∈ φ∗K(J(A)). Hence (φ τ ′)∗K(J(A)) = φ∗K(J(A)), so

(φ τ ′)∗k(J(A)) = φ∗k(J(A)).

By Proposition 2.5.28 there is a unique τ respecting polarizations such that

B1 ×B2φ //

φτ ′ $$

J(A)

τ

J(A)

commutes. Finally we see that since τ ′ is non-trivial so τ is also non-trivial.

We see by combining Corollary 4.1.1 with Torelli’s theorem that A must double cover a curve C.

Lemma 4.1.2. Let φ : B1 × B2 → J(A) be as before and let τ : J(A) → J(A) be the induced

involution. Then there is a non-trivial isomorphism α : A→ A such that α∗ = τ or α∗ = −τ .

Proof. Let P0 ∈ A(k) be a point and let j : A→ J(A) be the morphism

j(P ) = [P − P0]


as in Proposition 2.5.31. Note that j is not necessarily defined over k. By the Torelli theorem

(Theorem 2.5.32) there is a c ∈ J(A) and α : A→ A such that

jα = ±τj + c.

Without loss of generality we may assume the sign is positive. We see by evaluating both sides at

P0 that c = jα(P0). Let D ∈ j(A) and write D = [P − P0] = j(P ). Then

τD = τjP

= jα(P )− jα(P0)

= [α(P )− P0]− [α(P0)− P0]

= α∗D.

So τ = α∗ when restricted to j(A) and since elements of j(A) generate (as a group) J(A)(k) (See

[10, Theorem A.8.1.1]) we have τ = α∗. Finally, since τ is nontrivial α is also nontrivial.

The only thing left to verify is that g(C) = 2.

Lemma 4.1.3. Let φ : B1 × B2 → J(A) and τ be as before. Let α : A → A be the induced

involution and let C := A/α be the double covered curve. Then g(C) = 2.

Proof. Without loss of generality assume that τ = α∗ (so by abuse of notation we write τ = α).

Let π : A → C be the quotient map. Then as usual there is the induced morphisms of Jacobians

π∗ : J(C)→ J(A). Since π∗π∗ = [2] we see π∗ has finite kernel. Since 〈τ〉 = Aut(A/C) we have

by Lemma 3.5.2 that π∗J(C) = Im(1+τ). We also observe that

B1 ×B2φ //

1+τ ′

J(A)

1+τ

[2]B1φ // Im(1+τ)

commutes. But φ [2] has finite kernel. Hence since π∗ : J(C)→ Im(1+τ) and

φ [2] : B1 → Im(1+τ) are isogenies onto Im(1+τ) we see

dim J(C) = dim(B1) = 2.

So g(C) = dim(B1) = 2.


4.2 Other future directions

Let J2,2 be the set of genus 4 curves whose Jacobians are decomposable according to our restric-

tions. Since every genus 2 curve is hyperelliptic we can vary the admissible choices of Cf for

the norm construction (Proposition 3.4.5) across the whole family of genus 2 curvesM2. We can

also vary µ across the set of all choices of involutions, which we shall call Conf2 P1k. The norm

construction gives a map of sets given by polynomial equations

ϕ : U → J2,2

where U ⊆ M2 × Conf2 P1k is the subset of pairs satisfying the technical conditions of the norm

construction. We can ask how well φ classifies the objects in J2,2. We conjecture that

Conjecture 4.2.1. For each J ∈ Im(ϕ) the set ϕ−1(J) is finite.

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http://www.jmilne.org/math/CourseNotes/index.html

Appendix A

MAGMA script

A.1 "Elliptic_Decomposition.m"

//This code builds a pair of genus 2 curves that both cover two elliptic

//curves.

d := 14;

k := FiniteField(101);

_<t>:=PolynomialRing(k);

//This is our starting x-line on the bottom.

KPX<X>:=FunctionField(k);

_<TP1>:=PolynomialRing(KPX);

//This is our t-line constructed as an extension of KPX.

KPT<T> := ext<KPX|TP1^2 - X*TP1 + d>;

_<TP2>:=PolynomialRing(KPT);

//and this is our involution by design

muT := d/T;

64

APPENDIX A. MAGMA SCRIPT 65

//We verify that we have done things correctly so far.

assert Genus(KPT) eq 0;

assert X eq (T + muT);

//We give an f used to construct C_f.

f := ((t)^2-1)*((t)^2-4)*((t)^2-9);

fT := Evaluate(f,T);

KCf := ext<KPT| TP2^2 - fT>;

assert Genus(KCf) eq 2;

//We identify our invariant and orbiting bits

finv := 1/2*(Evaluate(f,t) + Evaluate(f,muT));

forb := 1/2*(Evaluate(f,T) - Evaluate(f,muT));

//Construct F, r(x) = finv

F := Norm(fT);

r := (1/2)*Trace(fT);

//We construct a hyperelliptic genus 2 curve from F

KCF<Y>:=ext<KPX|TP1^2-F>;

_<TCF>:=PolynomialRing(KCF);

assert Genus(KCF) eq 2;

//This is the genus 4 curve.

KD1<Z>:=ext<KCF|TCF^2-(r-Y)>;

assert Genus(KD1) eq 4;

//Here we build a model of the curve.

P3<x,w,y,z> := ProjectiveSpace(k,[1,1,3,3]);

D := Curve(P3, [ Numerator(y^2 - w^6*Evaluate(F,x/w)),

APPENDIX A. MAGMA SCRIPT 66

Numerator(z^2 - (w^6*Evaluate(g, x/w)) - w^3*y)]);

//Now we check the automorphism groups.

AutomorphismGroup(KCF);

AutomorphismGroup(KCf);

AutomorphismGroup(KD1);

//These all seem to be unusually large as expected.

assert IsHyperelliptic(D) eq false;

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